(0) Obligation:
Clauses:
rotate(X, Y) :- ','(append(A, B, X), append(B, A, Y)).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
append([], Ys, Ys).
Query: rotate(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, X4, .(X1, X5)) :- appendB(X2, X3, X4, X5).
appendC(.(X1, X2), .(X1, X3)) :- appendC(X2, X3).
rotateD(.(X1, X2), X3) :- appendA(X4, X5, X2).
rotateD(.(X1, X2), X3) :- ','(appendcA(X4, X5, X2), appendB(X5, X1, X4, X3)).
rotateD(X1, X2) :- appendC(X1, X2).
Clauses:
appendcA(.(X1, X2), X3, .(X1, X4)) :- appendcA(X2, X3, X4).
appendcA([], X1, X1).
appendcB(.(X1, X2), X3, X4, .(X1, X5)) :- appendcB(X2, X3, X4, X5).
appendcB([], X1, X2, .(X1, X2)).
appendcC(.(X1, X2), .(X1, X3)) :- appendcC(X2, X3).
appendcC([], []).
Afs:
rotateD(x1, x2) = rotateD(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:
rotateD_in: (b,f)
appendA_in: (f,f,b)
appendcA_in: (f,f,b)
appendB_in: (b,b,b,f)
appendC_in: (b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
ROTATED_IN_GA(.(X1, X2), X3) → U4_GA(X1, X2, X3, appendA_in_aag(X4, X5, X2))
ROTATED_IN_GA(.(X1, X2), X3) → APPENDA_IN_AAG(X4, X5, X2)
APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) → U1_AAG(X1, X2, X3, X4, appendA_in_aag(X2, X3, X4))
APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_AAG(X2, X3, X4)
ROTATED_IN_GA(.(X1, X2), X3) → U5_GA(X1, X2, X3, appendcA_in_aag(X4, X5, X2))
U5_GA(X1, X2, X3, appendcA_out_aag(X4, X5, X2)) → U6_GA(X1, X2, X3, appendB_in_ggga(X5, X1, X4, X3))
U5_GA(X1, X2, X3, appendcA_out_aag(X4, X5, X2)) → APPENDB_IN_GGGA(X5, X1, X4, X3)
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U2_GGGA(X1, X2, X3, X4, X5, appendB_in_ggga(X2, X3, X4, X5))
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)
ROTATED_IN_GA(X1, X2) → U7_GA(X1, X2, appendC_in_ga(X1, X2))
ROTATED_IN_GA(X1, X2) → APPENDC_IN_GA(X1, X2)
APPENDC_IN_GA(.(X1, X2), .(X1, X3)) → U3_GA(X1, X2, X3, appendC_in_ga(X2, X3))
APPENDC_IN_GA(.(X1, X2), .(X1, X3)) → APPENDC_IN_GA(X2, X3)
The TRS R consists of the following rules:
appendcA_in_aag(.(X1, X2), X3, .(X1, X4)) → U9_aag(X1, X2, X3, X4, appendcA_in_aag(X2, X3, X4))
appendcA_in_aag([], X1, X1) → appendcA_out_aag([], X1, X1)
U9_aag(X1, X2, X3, X4, appendcA_out_aag(X2, X3, X4)) → appendcA_out_aag(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
appendA_in_aag(
x1,
x2,
x3) =
appendA_in_aag(
x3)
appendcA_in_aag(
x1,
x2,
x3) =
appendcA_in_aag(
x3)
U9_aag(
x1,
x2,
x3,
x4,
x5) =
U9_aag(
x1,
x4,
x5)
appendcA_out_aag(
x1,
x2,
x3) =
appendcA_out_aag(
x1,
x2,
x3)
appendB_in_ggga(
x1,
x2,
x3,
x4) =
appendB_in_ggga(
x1,
x2,
x3)
appendC_in_ga(
x1,
x2) =
appendC_in_ga(
x1)
ROTATED_IN_GA(
x1,
x2) =
ROTATED_IN_GA(
x1)
U4_GA(
x1,
x2,
x3,
x4) =
U4_GA(
x1,
x2,
x4)
APPENDA_IN_AAG(
x1,
x2,
x3) =
APPENDA_IN_AAG(
x3)
U1_AAG(
x1,
x2,
x3,
x4,
x5) =
U1_AAG(
x1,
x4,
x5)
U5_GA(
x1,
x2,
x3,
x4) =
U5_GA(
x1,
x2,
x4)
U6_GA(
x1,
x2,
x3,
x4) =
U6_GA(
x1,
x2,
x4)
APPENDB_IN_GGGA(
x1,
x2,
x3,
x4) =
APPENDB_IN_GGGA(
x1,
x2,
x3)
U2_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GGGA(
x1,
x2,
x3,
x4,
x6)
U7_GA(
x1,
x2,
x3) =
U7_GA(
x1,
x3)
APPENDC_IN_GA(
x1,
x2) =
APPENDC_IN_GA(
x1)
U3_GA(
x1,
x2,
x3,
x4) =
U3_GA(
x1,
x2,
x4)
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:
ROTATED_IN_GA(.(X1, X2), X3) → U4_GA(X1, X2, X3, appendA_in_aag(X4, X5, X2))
ROTATED_IN_GA(.(X1, X2), X3) → APPENDA_IN_AAG(X4, X5, X2)
APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) → U1_AAG(X1, X2, X3, X4, appendA_in_aag(X2, X3, X4))
APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_AAG(X2, X3, X4)
ROTATED_IN_GA(.(X1, X2), X3) → U5_GA(X1, X2, X3, appendcA_in_aag(X4, X5, X2))
U5_GA(X1, X2, X3, appendcA_out_aag(X4, X5, X2)) → U6_GA(X1, X2, X3, appendB_in_ggga(X5, X1, X4, X3))
U5_GA(X1, X2, X3, appendcA_out_aag(X4, X5, X2)) → APPENDB_IN_GGGA(X5, X1, X4, X3)
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U2_GGGA(X1, X2, X3, X4, X5, appendB_in_ggga(X2, X3, X4, X5))
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)
ROTATED_IN_GA(X1, X2) → U7_GA(X1, X2, appendC_in_ga(X1, X2))
ROTATED_IN_GA(X1, X2) → APPENDC_IN_GA(X1, X2)
APPENDC_IN_GA(.(X1, X2), .(X1, X3)) → U3_GA(X1, X2, X3, appendC_in_ga(X2, X3))
APPENDC_IN_GA(.(X1, X2), .(X1, X3)) → APPENDC_IN_GA(X2, X3)
The TRS R consists of the following rules:
appendcA_in_aag(.(X1, X2), X3, .(X1, X4)) → U9_aag(X1, X2, X3, X4, appendcA_in_aag(X2, X3, X4))
appendcA_in_aag([], X1, X1) → appendcA_out_aag([], X1, X1)
U9_aag(X1, X2, X3, X4, appendcA_out_aag(X2, X3, X4)) → appendcA_out_aag(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
appendA_in_aag(
x1,
x2,
x3) =
appendA_in_aag(
x3)
appendcA_in_aag(
x1,
x2,
x3) =
appendcA_in_aag(
x3)
U9_aag(
x1,
x2,
x3,
x4,
x5) =
U9_aag(
x1,
x4,
x5)
appendcA_out_aag(
x1,
x2,
x3) =
appendcA_out_aag(
x1,
x2,
x3)
appendB_in_ggga(
x1,
x2,
x3,
x4) =
appendB_in_ggga(
x1,
x2,
x3)
appendC_in_ga(
x1,
x2) =
appendC_in_ga(
x1)
ROTATED_IN_GA(
x1,
x2) =
ROTATED_IN_GA(
x1)
U4_GA(
x1,
x2,
x3,
x4) =
U4_GA(
x1,
x2,
x4)
APPENDA_IN_AAG(
x1,
x2,
x3) =
APPENDA_IN_AAG(
x3)
U1_AAG(
x1,
x2,
x3,
x4,
x5) =
U1_AAG(
x1,
x4,
x5)
U5_GA(
x1,
x2,
x3,
x4) =
U5_GA(
x1,
x2,
x4)
U6_GA(
x1,
x2,
x3,
x4) =
U6_GA(
x1,
x2,
x4)
APPENDB_IN_GGGA(
x1,
x2,
x3,
x4) =
APPENDB_IN_GGGA(
x1,
x2,
x3)
U2_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GGGA(
x1,
x2,
x3,
x4,
x6)
U7_GA(
x1,
x2,
x3) =
U7_GA(
x1,
x3)
APPENDC_IN_GA(
x1,
x2) =
APPENDC_IN_GA(
x1)
U3_GA(
x1,
x2,
x3,
x4) =
U3_GA(
x1,
x2,
x4)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDC_IN_GA(.(X1, X2), .(X1, X3)) → APPENDC_IN_GA(X2, X3)
The TRS R consists of the following rules:
appendcA_in_aag(.(X1, X2), X3, .(X1, X4)) → U9_aag(X1, X2, X3, X4, appendcA_in_aag(X2, X3, X4))
appendcA_in_aag([], X1, X1) → appendcA_out_aag([], X1, X1)
U9_aag(X1, X2, X3, X4, appendcA_out_aag(X2, X3, X4)) → appendcA_out_aag(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
appendcA_in_aag(
x1,
x2,
x3) =
appendcA_in_aag(
x3)
U9_aag(
x1,
x2,
x3,
x4,
x5) =
U9_aag(
x1,
x4,
x5)
appendcA_out_aag(
x1,
x2,
x3) =
appendcA_out_aag(
x1,
x2,
x3)
APPENDC_IN_GA(
x1,
x2) =
APPENDC_IN_GA(
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:
APPENDC_IN_GA(.(X1, X2), .(X1, X3)) → APPENDC_IN_GA(X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDC_IN_GA(
x1,
x2) =
APPENDC_IN_GA(
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:
APPENDC_IN_GA(.(X1, X2)) → APPENDC_IN_GA(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:
- APPENDC_IN_GA(.(X1, X2)) → APPENDC_IN_GA(X2)
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)
The TRS R consists of the following rules:
appendcA_in_aag(.(X1, X2), X3, .(X1, X4)) → U9_aag(X1, X2, X3, X4, appendcA_in_aag(X2, X3, X4))
appendcA_in_aag([], X1, X1) → appendcA_out_aag([], X1, X1)
U9_aag(X1, X2, X3, X4, appendcA_out_aag(X2, X3, X4)) → appendcA_out_aag(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
appendcA_in_aag(
x1,
x2,
x3) =
appendcA_in_aag(
x3)
U9_aag(
x1,
x2,
x3,
x4,
x5) =
U9_aag(
x1,
x4,
x5)
appendcA_out_aag(
x1,
x2,
x3) =
appendcA_out_aag(
x1,
x2,
x3)
APPENDB_IN_GGGA(
x1,
x2,
x3,
x4) =
APPENDB_IN_GGGA(
x1,
x2,
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:
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDB_IN_GGGA(
x1,
x2,
x3,
x4) =
APPENDB_IN_GGGA(
x1,
x2,
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:
APPENDB_IN_GGGA(.(X1, X2), X3, X4) → APPENDB_IN_GGGA(X2, X3, X4)
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:
- APPENDB_IN_GGGA(.(X1, X2), X3, X4) → APPENDB_IN_GGGA(X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
(20) YES
(21) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_AAG(X2, X3, X4)
The TRS R consists of the following rules:
appendcA_in_aag(.(X1, X2), X3, .(X1, X4)) → U9_aag(X1, X2, X3, X4, appendcA_in_aag(X2, X3, X4))
appendcA_in_aag([], X1, X1) → appendcA_out_aag([], X1, X1)
U9_aag(X1, X2, X3, X4, appendcA_out_aag(X2, X3, X4)) → appendcA_out_aag(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
appendcA_in_aag(
x1,
x2,
x3) =
appendcA_in_aag(
x3)
U9_aag(
x1,
x2,
x3,
x4,
x5) =
U9_aag(
x1,
x4,
x5)
appendcA_out_aag(
x1,
x2,
x3) =
appendcA_out_aag(
x1,
x2,
x3)
APPENDA_IN_AAG(
x1,
x2,
x3) =
APPENDA_IN_AAG(
x3)
We have to consider all (P,R,Pi)-chains
(22) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(23) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_AAG(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDA_IN_AAG(
x1,
x2,
x3) =
APPENDA_IN_AAG(
x3)
We have to consider all (P,R,Pi)-chains
(24) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(25) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APPENDA_IN_AAG(.(X1, X4)) → APPENDA_IN_AAG(X4)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(26) 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_AAG(.(X1, X4)) → APPENDA_IN_AAG(X4)
The graph contains the following edges 1 > 1
(27) YES