(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)).
Query: goal(g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
s2lA(s(X1), .(X2, X3)) :- s2lA(X1, X3).
listB(.(X1, X2)) :- listB(X2).
goalC(s(X1)) :- s2lA(X1, X2).
goalC(s(X1)) :- ','(s2lcA(X1, X2), listB(X2)).
Clauses:
s2lcA(s(X1), .(X2, X3)) :- s2lcA(X1, X3).
s2lcA(0, []).
listcB([]).
listcB(.(X1, X2)) :- listcB(X2).
Afs:
goalC(x1) = goalC(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
goalC_in: (b)
s2lA_in: (b,f)
s2lcA_in: (b,f)
listB_in: (b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
GOALC_IN_G(s(X1)) → U3_G(X1, s2lA_in_ga(X1, X2))
GOALC_IN_G(s(X1)) → S2LA_IN_GA(X1, X2)
S2LA_IN_GA(s(X1), .(X2, X3)) → U1_GA(X1, X2, X3, s2lA_in_ga(X1, X3))
S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)
GOALC_IN_G(s(X1)) → U4_G(X1, s2lcA_in_ga(X1, X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → U5_G(X1, listB_in_g(X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → LISTB_IN_G(X2)
LISTB_IN_G(.(X1, X2)) → U2_G(X1, X2, listB_in_g(X2))
LISTB_IN_G(.(X1, X2)) → LISTB_IN_G(X2)
The TRS R consists of the following rules:
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
s2lA_in_ga(
x1,
x2) =
s2lA_in_ga(
x1)
.(
x1,
x2) =
.(
x2)
s2lcA_in_ga(
x1,
x2) =
s2lcA_in_ga(
x1)
U7_ga(
x1,
x2,
x3,
x4) =
U7_ga(
x1,
x4)
0 =
0
s2lcA_out_ga(
x1,
x2) =
s2lcA_out_ga(
x1,
x2)
listB_in_g(
x1) =
listB_in_g(
x1)
GOALC_IN_G(
x1) =
GOALC_IN_G(
x1)
U3_G(
x1,
x2) =
U3_G(
x1,
x2)
S2LA_IN_GA(
x1,
x2) =
S2LA_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)
LISTB_IN_G(
x1) =
LISTB_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:
GOALC_IN_G(s(X1)) → U3_G(X1, s2lA_in_ga(X1, X2))
GOALC_IN_G(s(X1)) → S2LA_IN_GA(X1, X2)
S2LA_IN_GA(s(X1), .(X2, X3)) → U1_GA(X1, X2, X3, s2lA_in_ga(X1, X3))
S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)
GOALC_IN_G(s(X1)) → U4_G(X1, s2lcA_in_ga(X1, X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → U5_G(X1, listB_in_g(X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → LISTB_IN_G(X2)
LISTB_IN_G(.(X1, X2)) → U2_G(X1, X2, listB_in_g(X2))
LISTB_IN_G(.(X1, X2)) → LISTB_IN_G(X2)
The TRS R consists of the following rules:
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
s2lA_in_ga(
x1,
x2) =
s2lA_in_ga(
x1)
.(
x1,
x2) =
.(
x2)
s2lcA_in_ga(
x1,
x2) =
s2lcA_in_ga(
x1)
U7_ga(
x1,
x2,
x3,
x4) =
U7_ga(
x1,
x4)
0 =
0
s2lcA_out_ga(
x1,
x2) =
s2lcA_out_ga(
x1,
x2)
listB_in_g(
x1) =
listB_in_g(
x1)
GOALC_IN_G(
x1) =
GOALC_IN_G(
x1)
U3_G(
x1,
x2) =
U3_G(
x1,
x2)
S2LA_IN_GA(
x1,
x2) =
S2LA_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)
LISTB_IN_G(
x1) =
LISTB_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:
LISTB_IN_G(.(X1, X2)) → LISTB_IN_G(X2)
The TRS R consists of the following rules:
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
.(
x1,
x2) =
.(
x2)
s2lcA_in_ga(
x1,
x2) =
s2lcA_in_ga(
x1)
U7_ga(
x1,
x2,
x3,
x4) =
U7_ga(
x1,
x4)
0 =
0
s2lcA_out_ga(
x1,
x2) =
s2lcA_out_ga(
x1,
x2)
LISTB_IN_G(
x1) =
LISTB_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:
LISTB_IN_G(.(X1, X2)) → LISTB_IN_G(X2)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x2)
LISTB_IN_G(
x1) =
LISTB_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:
LISTB_IN_G(.(X2)) → LISTB_IN_G(X2)
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:
- LISTB_IN_G(.(X2)) → LISTB_IN_G(X2)
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)
The TRS R consists of the following rules:
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
.(
x1,
x2) =
.(
x2)
s2lcA_in_ga(
x1,
x2) =
s2lcA_in_ga(
x1)
U7_ga(
x1,
x2,
x3,
x4) =
U7_ga(
x1,
x4)
0 =
0
s2lcA_out_ga(
x1,
x2) =
s2lcA_out_ga(
x1,
x2)
S2LA_IN_GA(
x1,
x2) =
S2LA_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:
S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
.(
x1,
x2) =
.(
x2)
S2LA_IN_GA(
x1,
x2) =
S2LA_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:
S2LA_IN_GA(s(X1)) → S2LA_IN_GA(X1)
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:
- S2LA_IN_GA(s(X1)) → S2LA_IN_GA(X1)
The graph contains the following edges 1 > 1
(20) YES