(0) Obligation:
Clauses:append([], XS, XS).
append(.(X, XS), YS, .(X, ZS)) :- append(XS, YS, ZS).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), append(XS, YS, ZS)).
Queries:
goal(g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:s2l9(s(T16), .(X67, X68)) :- s2l9(T16, X68).
append21(.(T28, T30), X143, .(T28, X144)) :- append21(T30, X143, X144).
goal1(s(T8)) :- s2l9(T8, X30).
goal1(s(T8)) :- ','(s2lc9(T8, T23), append21(T23, X110, X111)).
Clauses:
s2lc9(s(T16), .(X67, X68)) :- s2lc9(T16, X68).
s2lc9(0, []).
appendc21([], X125, X125).
appendc21(.(T28, T30), X143, .(T28, X144)) :- appendc21(T30, X143, X144).
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)
append21_in: (b,f,f)
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, X30))
GOAL1_IN_G(s(T8)) → S2L9_IN_GA(T8, X30)
S2L9_IN_GA(s(T16), .(X67, X68)) → U1_GA(T16, X67, X68, s2l9_in_ga(T16, X68))
S2L9_IN_GA(s(T16), .(X67, X68)) → S2L9_IN_GA(T16, X68)
GOAL1_IN_G(s(T8)) → U4_G(T8, s2lc9_in_ga(T8, T23))
U4_G(T8, s2lc9_out_ga(T8, T23)) → U5_G(T8, append21_in_gaa(T23, X110, X111))
U4_G(T8, s2lc9_out_ga(T8, T23)) → APPEND21_IN_GAA(T23, X110, X111)
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → U2_GAA(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → APPEND21_IN_GAA(T30, X143, X144)
The TRS R consists of the following rules:
s2lc9_in_ga(s(T16), .(X67, X68)) → U7_ga(T16, X67, X68, s2lc9_in_ga(T16, X68))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U7_ga(T16, X67, X68, s2lc9_out_ga(T16, X68)) → s2lc9_out_ga(s(T16), .(X67, X68))
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)
append21_in_gaa(x1, x2, x3) = append21_in_gaa(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)
APPEND21_IN_GAA(x1, x2, x3) = APPEND21_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x2, x5)
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, X30))
GOAL1_IN_G(s(T8)) → S2L9_IN_GA(T8, X30)
S2L9_IN_GA(s(T16), .(X67, X68)) → U1_GA(T16, X67, X68, s2l9_in_ga(T16, X68))
S2L9_IN_GA(s(T16), .(X67, X68)) → S2L9_IN_GA(T16, X68)
GOAL1_IN_G(s(T8)) → U4_G(T8, s2lc9_in_ga(T8, T23))
U4_G(T8, s2lc9_out_ga(T8, T23)) → U5_G(T8, append21_in_gaa(T23, X110, X111))
U4_G(T8, s2lc9_out_ga(T8, T23)) → APPEND21_IN_GAA(T23, X110, X111)
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → U2_GAA(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → APPEND21_IN_GAA(T30, X143, X144)
The TRS R consists of the following rules:
s2lc9_in_ga(s(T16), .(X67, X68)) → U7_ga(T16, X67, X68, s2lc9_in_ga(T16, X68))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U7_ga(T16, X67, X68, s2lc9_out_ga(T16, X68)) → s2lc9_out_ga(s(T16), .(X67, X68))
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)
append21_in_gaa(x1, x2, x3) = append21_in_gaa(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)
APPEND21_IN_GAA(x1, x2, x3) = APPEND21_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x2, x5)
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:
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → APPEND21_IN_GAA(T30, X143, X144)
The TRS R consists of the following rules:
s2lc9_in_ga(s(T16), .(X67, X68)) → U7_ga(T16, X67, X68, s2lc9_in_ga(T16, X68))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U7_ga(T16, X67, X68, s2lc9_out_ga(T16, X68)) → s2lc9_out_ga(s(T16), .(X67, X68))
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)
APPEND21_IN_GAA(x1, x2, x3) = APPEND21_IN_GAA(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:
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → APPEND21_IN_GAA(T30, X143, X144)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APPEND21_IN_GAA(x1, x2, x3) = APPEND21_IN_GAA(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:
APPEND21_IN_GAA(.(T30)) → APPEND21_IN_GAA(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:
- APPEND21_IN_GAA(.(T30)) → APPEND21_IN_GAA(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), .(X67, X68)) → S2L9_IN_GA(T16, X68)
The TRS R consists of the following rules:
s2lc9_in_ga(s(T16), .(X67, X68)) → U7_ga(T16, X67, X68, s2lc9_in_ga(T16, X68))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U7_ga(T16, X67, X68, s2lc9_out_ga(T16, X68)) → s2lc9_out_ga(s(T16), .(X67, X68))
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), .(X67, X68)) → S2L9_IN_GA(T16, X68)
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