(0) Obligation:
Clauses:
append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)).
Query: append3(g,g,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
appendA(.(X1, X2), X3, .(X1, X4)) :- appendA(X2, X3, X4).
appendB(.(X1, X2), X3, .(X1, X4)) :- appendB(X2, X3, X4).
append3C([], X1, X2, X3) :- appendA(X1, X2, X3).
append3C(.(X1, X2), X3, X4, X5) :- appendB(X2, X3, X6).
append3C(.(X1, X2), X3, X4, X5) :- ','(appendcB(X2, X3, X6), appendA(.(X1, X6), X4, X5)).
Clauses:
appendcA([], X1, X1).
appendcA(.(X1, X2), X3, .(X1, X4)) :- appendcA(X2, X3, X4).
appendcB([], X1, X1).
appendcB(.(X1, X2), X3, .(X1, X4)) :- appendcB(X2, X3, X4).
Afs:
append3C(x1, x2, x3, x4) = append3C(x1, x2, x3)
(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:
append3C_in: (b,b,b,f)
appendA_in: (b,b,f)
appendB_in: (b,b,f)
appendcB_in: (b,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APPEND3C_IN_GGGA([], X1, X2, X3) → U3_GGGA(X1, X2, X3, appendA_in_gga(X1, X2, X3))
APPEND3C_IN_GGGA([], X1, X2, X3) → APPENDA_IN_GGA(X1, X2, X3)
APPENDA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U1_GGA(X1, X2, X3, X4, appendA_in_gga(X2, X3, X4))
APPENDA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GGA(X2, X3, X4)
APPEND3C_IN_GGGA(.(X1, X2), X3, X4, X5) → U4_GGGA(X1, X2, X3, X4, X5, appendB_in_gga(X2, X3, X6))
APPEND3C_IN_GGGA(.(X1, X2), X3, X4, X5) → APPENDB_IN_GGA(X2, X3, X6)
APPENDB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U2_GGA(X1, X2, X3, X4, appendB_in_gga(X2, X3, X4))
APPENDB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPENDB_IN_GGA(X2, X3, X4)
APPEND3C_IN_GGGA(.(X1, X2), X3, X4, X5) → U5_GGGA(X1, X2, X3, X4, X5, appendcB_in_gga(X2, X3, X6))
U5_GGGA(X1, X2, X3, X4, X5, appendcB_out_gga(X2, X3, X6)) → U6_GGGA(X1, X2, X3, X4, X5, appendA_in_gga(.(X1, X6), X4, X5))
U5_GGGA(X1, X2, X3, X4, X5, appendcB_out_gga(X2, X3, X6)) → APPENDA_IN_GGA(.(X1, X6), X4, X5)
The TRS R consists of the following rules:
appendcB_in_gga([], X1, X1) → appendcB_out_gga([], X1, X1)
appendcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U9_gga(X1, X2, X3, X4, appendcB_in_gga(X2, X3, X4))
U9_gga(X1, X2, X3, X4, appendcB_out_gga(X2, X3, X4)) → appendcB_out_gga(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
[] =
[]
appendA_in_gga(
x1,
x2,
x3) =
appendA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
appendB_in_gga(
x1,
x2,
x3) =
appendB_in_gga(
x1,
x2)
appendcB_in_gga(
x1,
x2,
x3) =
appendcB_in_gga(
x1,
x2)
appendcB_out_gga(
x1,
x2,
x3) =
appendcB_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x3,
x5)
APPEND3C_IN_GGGA(
x1,
x2,
x3,
x4) =
APPEND3C_IN_GGGA(
x1,
x2,
x3)
U3_GGGA(
x1,
x2,
x3,
x4) =
U3_GGGA(
x1,
x2,
x4)
APPENDA_IN_GGA(
x1,
x2,
x3) =
APPENDA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGA(
x1,
x2,
x3,
x5)
U4_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_GGGA(
x1,
x2,
x3,
x4,
x6)
APPENDB_IN_GGA(
x1,
x2,
x3) =
APPENDB_IN_GGA(
x1,
x2)
U2_GGA(
x1,
x2,
x3,
x4,
x5) =
U2_GGA(
x1,
x2,
x3,
x5)
U5_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U5_GGGA(
x1,
x2,
x3,
x4,
x6)
U6_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_GGGA(
x1,
x2,
x3,
x4,
x6)
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:
APPEND3C_IN_GGGA([], X1, X2, X3) → U3_GGGA(X1, X2, X3, appendA_in_gga(X1, X2, X3))
APPEND3C_IN_GGGA([], X1, X2, X3) → APPENDA_IN_GGA(X1, X2, X3)
APPENDA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U1_GGA(X1, X2, X3, X4, appendA_in_gga(X2, X3, X4))
APPENDA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GGA(X2, X3, X4)
APPEND3C_IN_GGGA(.(X1, X2), X3, X4, X5) → U4_GGGA(X1, X2, X3, X4, X5, appendB_in_gga(X2, X3, X6))
APPEND3C_IN_GGGA(.(X1, X2), X3, X4, X5) → APPENDB_IN_GGA(X2, X3, X6)
APPENDB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U2_GGA(X1, X2, X3, X4, appendB_in_gga(X2, X3, X4))
APPENDB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPENDB_IN_GGA(X2, X3, X4)
APPEND3C_IN_GGGA(.(X1, X2), X3, X4, X5) → U5_GGGA(X1, X2, X3, X4, X5, appendcB_in_gga(X2, X3, X6))
U5_GGGA(X1, X2, X3, X4, X5, appendcB_out_gga(X2, X3, X6)) → U6_GGGA(X1, X2, X3, X4, X5, appendA_in_gga(.(X1, X6), X4, X5))
U5_GGGA(X1, X2, X3, X4, X5, appendcB_out_gga(X2, X3, X6)) → APPENDA_IN_GGA(.(X1, X6), X4, X5)
The TRS R consists of the following rules:
appendcB_in_gga([], X1, X1) → appendcB_out_gga([], X1, X1)
appendcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U9_gga(X1, X2, X3, X4, appendcB_in_gga(X2, X3, X4))
U9_gga(X1, X2, X3, X4, appendcB_out_gga(X2, X3, X4)) → appendcB_out_gga(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
[] =
[]
appendA_in_gga(
x1,
x2,
x3) =
appendA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
appendB_in_gga(
x1,
x2,
x3) =
appendB_in_gga(
x1,
x2)
appendcB_in_gga(
x1,
x2,
x3) =
appendcB_in_gga(
x1,
x2)
appendcB_out_gga(
x1,
x2,
x3) =
appendcB_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x3,
x5)
APPEND3C_IN_GGGA(
x1,
x2,
x3,
x4) =
APPEND3C_IN_GGGA(
x1,
x2,
x3)
U3_GGGA(
x1,
x2,
x3,
x4) =
U3_GGGA(
x1,
x2,
x4)
APPENDA_IN_GGA(
x1,
x2,
x3) =
APPENDA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGA(
x1,
x2,
x3,
x5)
U4_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_GGGA(
x1,
x2,
x3,
x4,
x6)
APPENDB_IN_GGA(
x1,
x2,
x3) =
APPENDB_IN_GGA(
x1,
x2)
U2_GGA(
x1,
x2,
x3,
x4,
x5) =
U2_GGA(
x1,
x2,
x3,
x5)
U5_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U5_GGGA(
x1,
x2,
x3,
x4,
x6)
U6_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_GGGA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPENDB_IN_GGA(X2, X3, X4)
The TRS R consists of the following rules:
appendcB_in_gga([], X1, X1) → appendcB_out_gga([], X1, X1)
appendcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U9_gga(X1, X2, X3, X4, appendcB_in_gga(X2, X3, X4))
U9_gga(X1, X2, X3, X4, appendcB_out_gga(X2, X3, X4)) → appendcB_out_gga(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x1,
x2)
appendcB_in_gga(
x1,
x2,
x3) =
appendcB_in_gga(
x1,
x2)
appendcB_out_gga(
x1,
x2,
x3) =
appendcB_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x3,
x5)
APPENDB_IN_GGA(
x1,
x2,
x3) =
APPENDB_IN_GGA(
x1,
x2)
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:
APPENDB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPENDB_IN_GGA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDB_IN_GGA(
x1,
x2,
x3) =
APPENDB_IN_GGA(
x1,
x2)
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:
APPENDB_IN_GGA(.(X1, X2), X3) → APPENDB_IN_GGA(X2, X3)
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:
- APPENDB_IN_GGA(.(X1, X2), X3) → APPENDB_IN_GGA(X2, X3)
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:
APPENDA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GGA(X2, X3, X4)
The TRS R consists of the following rules:
appendcB_in_gga([], X1, X1) → appendcB_out_gga([], X1, X1)
appendcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U9_gga(X1, X2, X3, X4, appendcB_in_gga(X2, X3, X4))
U9_gga(X1, X2, X3, X4, appendcB_out_gga(X2, X3, X4)) → appendcB_out_gga(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x1,
x2)
appendcB_in_gga(
x1,
x2,
x3) =
appendcB_in_gga(
x1,
x2)
appendcB_out_gga(
x1,
x2,
x3) =
appendcB_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x3,
x5)
APPENDA_IN_GGA(
x1,
x2,
x3) =
APPENDA_IN_GGA(
x1,
x2)
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:
APPENDA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GGA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDA_IN_GGA(
x1,
x2,
x3) =
APPENDA_IN_GGA(
x1,
x2)
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:
APPENDA_IN_GGA(.(X1, X2), X3) → APPENDA_IN_GGA(X2, X3)
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:
- APPENDA_IN_GGA(.(X1, X2), X3) → APPENDA_IN_GGA(X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2
(20) YES