(0) Obligation:
Clauses:append1([], Ys, Ys).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append2([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
sublist(X, Y) :- ','(append1(P, X1, Y), append2(X2, X, P)).
Queries:
sublist(g,g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:append117(.(X97, X98), X99, .(X97, T44)) :- append117(X98, X99, T44).
append232(.(X158, X159), T86, .(X158, T87)) :- append232(X159, T86, T87).
sublist1(T5, .(X56, T26)) :- append117(X57, X58, T26).
sublist1(T73, .(X132, T26)) :- ','(append1c17(T74, T35, T26), append232(X133, T73, T74)).
Clauses:
append1c17([], T41, T41).
append1c17(.(X97, X98), X99, .(X97, T44)) :- append1c17(X98, X99, T44).
append2c32([], T81, T81).
append2c32(.(X158, X159), T86, .(X158, T87)) :- append2c32(X159, T86, T87).
Afs:
sublist1(x1, x2) = sublist1(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:sublist1_in: (b,b)
append117_in: (f,f,b)
append1c17_in: (f,f,b)
append232_in: (f,b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SUBLIST1_IN_GG(T5, .(X56, T26)) → U3_GG(T5, X56, T26, append117_in_aag(X57, X58, T26))
SUBLIST1_IN_GG(T5, .(X56, T26)) → APPEND117_IN_AAG(X57, X58, T26)
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → U1_AAG(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_AAG(X98, X99, T44)
SUBLIST1_IN_GG(T73, .(X132, T26)) → U4_GG(T73, X132, T26, append1c17_in_aag(T74, T35, T26))
U4_GG(T73, X132, T26, append1c17_out_aag(T74, T35, T26)) → U5_GG(T73, X132, T26, append232_in_agg(X133, T73, T74))
U4_GG(T73, X132, T26, append1c17_out_aag(T74, T35, T26)) → APPEND232_IN_AGG(X133, T73, T74)
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → U2_AGG(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → APPEND232_IN_AGG(X159, T86, T87)
The TRS R consists of the following rules:
append1c17_in_aag([], T41, T41) → append1c17_out_aag([], T41, T41)
append1c17_in_aag(.(X97, X98), X99, .(X97, T44)) → U7_aag(X97, X98, X99, T44, append1c17_in_aag(X98, X99, T44))
U7_aag(X97, X98, X99, T44, append1c17_out_aag(X98, X99, T44)) → append1c17_out_aag(.(X97, X98), X99, .(X97, T44))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append1c17_in_aag(x1, x2, x3) = append1c17_in_aag(x3)
append1c17_out_aag(x1, x2, x3) = append1c17_out_aag(x1, x2, x3)
U7_aag(x1, x2, x3, x4, x5) = U7_aag(x1, x4, x5)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
SUBLIST1_IN_GG(x1, x2) = SUBLIST1_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4)
APPEND117_IN_AAG(x1, x2, x3) = APPEND117_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x4, x5)
U4_GG(x1, x2, x3, x4) = U4_GG(x1, x2, x3, x4)
U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4)
APPEND232_IN_AGG(x1, x2, x3) = APPEND232_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x1, x3, x4, 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:
SUBLIST1_IN_GG(T5, .(X56, T26)) → U3_GG(T5, X56, T26, append117_in_aag(X57, X58, T26))
SUBLIST1_IN_GG(T5, .(X56, T26)) → APPEND117_IN_AAG(X57, X58, T26)
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → U1_AAG(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_AAG(X98, X99, T44)
SUBLIST1_IN_GG(T73, .(X132, T26)) → U4_GG(T73, X132, T26, append1c17_in_aag(T74, T35, T26))
U4_GG(T73, X132, T26, append1c17_out_aag(T74, T35, T26)) → U5_GG(T73, X132, T26, append232_in_agg(X133, T73, T74))
U4_GG(T73, X132, T26, append1c17_out_aag(T74, T35, T26)) → APPEND232_IN_AGG(X133, T73, T74)
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → U2_AGG(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → APPEND232_IN_AGG(X159, T86, T87)
The TRS R consists of the following rules:
append1c17_in_aag([], T41, T41) → append1c17_out_aag([], T41, T41)
append1c17_in_aag(.(X97, X98), X99, .(X97, T44)) → U7_aag(X97, X98, X99, T44, append1c17_in_aag(X98, X99, T44))
U7_aag(X97, X98, X99, T44, append1c17_out_aag(X98, X99, T44)) → append1c17_out_aag(.(X97, X98), X99, .(X97, T44))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append1c17_in_aag(x1, x2, x3) = append1c17_in_aag(x3)
append1c17_out_aag(x1, x2, x3) = append1c17_out_aag(x1, x2, x3)
U7_aag(x1, x2, x3, x4, x5) = U7_aag(x1, x4, x5)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
SUBLIST1_IN_GG(x1, x2) = SUBLIST1_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4)
APPEND117_IN_AAG(x1, x2, x3) = APPEND117_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x4, x5)
U4_GG(x1, x2, x3, x4) = U4_GG(x1, x2, x3, x4)
U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4)
APPEND232_IN_AGG(x1, x2, x3) = APPEND232_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x1, x3, x4, 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:
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → APPEND232_IN_AGG(X159, T86, T87)
The TRS R consists of the following rules:
append1c17_in_aag([], T41, T41) → append1c17_out_aag([], T41, T41)
append1c17_in_aag(.(X97, X98), X99, .(X97, T44)) → U7_aag(X97, X98, X99, T44, append1c17_in_aag(X98, X99, T44))
U7_aag(X97, X98, X99, T44, append1c17_out_aag(X98, X99, T44)) → append1c17_out_aag(.(X97, X98), X99, .(X97, T44))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
append1c17_in_aag(x1, x2, x3) = append1c17_in_aag(x3)
append1c17_out_aag(x1, x2, x3) = append1c17_out_aag(x1, x2, x3)
U7_aag(x1, x2, x3, x4, x5) = U7_aag(x1, x4, x5)
APPEND232_IN_AGG(x1, x2, x3) = APPEND232_IN_AGG(x2, x3)
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:
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → APPEND232_IN_AGG(X159, T86, T87)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND232_IN_AGG(x1, x2, x3) = APPEND232_IN_AGG(x2, x3)
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:
APPEND232_IN_AGG(T86, .(X158, T87)) → APPEND232_IN_AGG(T86, T87)
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:
- APPEND232_IN_AGG(T86, .(X158, T87)) → APPEND232_IN_AGG(T86, T87)
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:
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_AAG(X98, X99, T44)
The TRS R consists of the following rules:
append1c17_in_aag([], T41, T41) → append1c17_out_aag([], T41, T41)
append1c17_in_aag(.(X97, X98), X99, .(X97, T44)) → U7_aag(X97, X98, X99, T44, append1c17_in_aag(X98, X99, T44))
U7_aag(X97, X98, X99, T44, append1c17_out_aag(X98, X99, T44)) → append1c17_out_aag(.(X97, X98), X99, .(X97, T44))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
append1c17_in_aag(x1, x2, x3) = append1c17_in_aag(x3)
append1c17_out_aag(x1, x2, x3) = append1c17_out_aag(x1, x2, x3)
U7_aag(x1, x2, x3, x4, x5) = U7_aag(x1, x4, x5)
APPEND117_IN_AAG(x1, x2, x3) = APPEND117_IN_AAG(x3)
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:
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_AAG(X98, X99, T44)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND117_IN_AAG(x1, x2, x3) = APPEND117_IN_AAG(x3)
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:
APPEND117_IN_AAG(.(X97, T44)) → APPEND117_IN_AAG(T44)
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:
- APPEND117_IN_AAG(.(X97, T44)) → APPEND117_IN_AAG(T44)
The graph contains the following edges 1 > 1