Left Termination of the query pattern
goal(g,a,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 88 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 18 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 5 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇒, 0 ms)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔, 0 ms)
        ↳27 YES

(0) Obligation:

Clauses:

goal(A, B, C) :- ','(s2l(A, D), applast(D, B, C)).
applast(L, X, Last) :- ','(append(L, .(X, []), LX), last(Last, LX)).
last(X, .(X, [])).
last(X, .(H, T)) :- last(X, T).
append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).

Query: goal(g,a,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

s2lA(s(X1), .(X2, X3)) :- s2lA(X1, X3).
appendB(.(X1, X2), X3, .(X1, X4)) :- appendB(X2, X3, X4).
lastC(X1, .(X2, X3)) :- lastC(X1, X3).
goalF(s(X1), X2, X3) :- s2lA(X1, X4).
goalF(s(X1), X2, X3) :- ','(s2lcA(X1, X4), appendB(X4, X2, X5)).
goalF(s(X1), X2, X3) :- ','(s2lcA(X1, X4), ','(appendcD(X5, X4, X2, X6), lastC(X3, X6))).
goalF(0, X1, X2) :- ','(appendcE(X1, X3), lastC(X2, X3)).

Clauses:

s2lcA(s(X1), .(X2, X3)) :- s2lcA(X1, X3).
s2lcA(0, []).
appendcB([], X1, .(X1, [])).
appendcB(.(X1, X2), X3, .(X1, X4)) :- appendcB(X2, X3, X4).
lastcC(X1, .(X1, [])).
lastcC(X1, .(X2, X3)) :- lastcC(X1, X3).
appendcD(X1, X2, X3, .(X1, X4)) :- appendcB(X2, X3, X4).
appendcE(X1, .(X1, [])).

Afs:

goalF(x1, x2, x3)  =  goalF(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:
goalF_in: (b,f,f)
s2lA_in: (b,f)
s2lcA_in: (b,f)
appendB_in: (b,f,f)
appendcD_in: (f,b,f,f)
appendcB_in: (b,f,f)
lastC_in: (f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOALF_IN_GAA(s(X1), X2, X3) → U4_GAA(X1, X2, X3, s2lA_in_ga(X1, X4))
GOALF_IN_GAA(s(X1), X2, X3) → S2LA_IN_GA(X1, X4)
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)
GOALF_IN_GAA(s(X1), X2, X3) → U5_GAA(X1, X2, X3, s2lcA_in_ga(X1, X4))
U5_GAA(X1, X2, X3, s2lcA_out_ga(X1, X4)) → U6_GAA(X1, X2, X3, appendB_in_gaa(X4, X2, X5))
U5_GAA(X1, X2, X3, s2lcA_out_ga(X1, X4)) → APPENDB_IN_GAA(X4, X2, X5)
APPENDB_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U2_GAA(X1, X2, X3, X4, appendB_in_gaa(X2, X3, X4))
APPENDB_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDB_IN_GAA(X2, X3, X4)
U5_GAA(X1, X2, X3, s2lcA_out_ga(X1, X4)) → U7_GAA(X1, X2, X3, appendcD_in_agaa(X5, X4, X2, X6))
U7_GAA(X1, X2, X3, appendcD_out_agaa(X5, X4, X2, X6)) → U8_GAA(X1, X2, X3, lastC_in_ag(X3, X6))
U7_GAA(X1, X2, X3, appendcD_out_agaa(X5, X4, X2, X6)) → LASTC_IN_AG(X3, X6)
LASTC_IN_AG(X1, .(X2, X3)) → U3_AG(X1, X2, X3, lastC_in_ag(X1, X3))
LASTC_IN_AG(X1, .(X2, X3)) → LASTC_IN_AG(X1, X3)
GOALF_IN_GAA(0, X1, X2) → U9_GAA(X1, X2, appendcE_in_aa(X1, X3))
U9_GAA(X1, X2, appendcE_out_aa(X1, X3)) → U10_GAA(X1, X2, lastC_in_ag(X2, X3))
U9_GAA(X1, X2, appendcE_out_aa(X1, X3)) → LASTC_IN_AG(X2, X3)

The TRS R consists of the following rules:

s2lcA_in_ga(s(X1), .(X2, X3)) → U12_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U12_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
appendcD_in_agaa(X1, X2, X3, .(X1, X4)) → U15_agaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
appendcB_in_gaa([], X1, .(X1, [])) → appendcB_out_gaa([], X1, .(X1, []))
appendcB_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcB_out_gaa(.(X1, X2), X3, .(X1, X4))
U15_agaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcD_out_agaa(X1, X2, X3, .(X1, X4))
appendcE_in_aa(X1, .(X1, [])) → appendcE_out_aa(X1, .(X1, []))

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)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
appendcD_in_agaa(x1, x2, x3, x4)  =  appendcD_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5)  =  U15_agaa(x2, x5)
appendcB_in_gaa(x1, x2, x3)  =  appendcB_in_gaa(x1)
[]  =  []
appendcB_out_gaa(x1, x2, x3)  =  appendcB_out_gaa(x1, x3)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
appendcD_out_agaa(x1, x2, x3, x4)  =  appendcD_out_agaa(x2, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
appendcE_in_aa(x1, x2)  =  appendcE_in_aa
appendcE_out_aa(x1, x2)  =  appendcE_out_aa(x2)
GOALF_IN_GAA(x1, x2, x3)  =  GOALF_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
APPENDB_IN_GAA(x1, x2, x3)  =  APPENDB_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
LASTC_IN_AG(x1, x2)  =  LASTC_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U9_GAA(x1, x2, x3)  =  U9_GAA(x3)
U10_GAA(x1, x2, x3)  =  U10_GAA(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:

GOALF_IN_GAA(s(X1), X2, X3) → U4_GAA(X1, X2, X3, s2lA_in_ga(X1, X4))
GOALF_IN_GAA(s(X1), X2, X3) → S2LA_IN_GA(X1, X4)
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)
GOALF_IN_GAA(s(X1), X2, X3) → U5_GAA(X1, X2, X3, s2lcA_in_ga(X1, X4))
U5_GAA(X1, X2, X3, s2lcA_out_ga(X1, X4)) → U6_GAA(X1, X2, X3, appendB_in_gaa(X4, X2, X5))
U5_GAA(X1, X2, X3, s2lcA_out_ga(X1, X4)) → APPENDB_IN_GAA(X4, X2, X5)
APPENDB_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U2_GAA(X1, X2, X3, X4, appendB_in_gaa(X2, X3, X4))
APPENDB_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDB_IN_GAA(X2, X3, X4)
U5_GAA(X1, X2, X3, s2lcA_out_ga(X1, X4)) → U7_GAA(X1, X2, X3, appendcD_in_agaa(X5, X4, X2, X6))
U7_GAA(X1, X2, X3, appendcD_out_agaa(X5, X4, X2, X6)) → U8_GAA(X1, X2, X3, lastC_in_ag(X3, X6))
U7_GAA(X1, X2, X3, appendcD_out_agaa(X5, X4, X2, X6)) → LASTC_IN_AG(X3, X6)
LASTC_IN_AG(X1, .(X2, X3)) → U3_AG(X1, X2, X3, lastC_in_ag(X1, X3))
LASTC_IN_AG(X1, .(X2, X3)) → LASTC_IN_AG(X1, X3)
GOALF_IN_GAA(0, X1, X2) → U9_GAA(X1, X2, appendcE_in_aa(X1, X3))
U9_GAA(X1, X2, appendcE_out_aa(X1, X3)) → U10_GAA(X1, X2, lastC_in_ag(X2, X3))
U9_GAA(X1, X2, appendcE_out_aa(X1, X3)) → LASTC_IN_AG(X2, X3)

The TRS R consists of the following rules:

s2lcA_in_ga(s(X1), .(X2, X3)) → U12_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U12_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
appendcD_in_agaa(X1, X2, X3, .(X1, X4)) → U15_agaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
appendcB_in_gaa([], X1, .(X1, [])) → appendcB_out_gaa([], X1, .(X1, []))
appendcB_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcB_out_gaa(.(X1, X2), X3, .(X1, X4))
U15_agaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcD_out_agaa(X1, X2, X3, .(X1, X4))
appendcE_in_aa(X1, .(X1, [])) → appendcE_out_aa(X1, .(X1, []))

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)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
appendcD_in_agaa(x1, x2, x3, x4)  =  appendcD_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5)  =  U15_agaa(x2, x5)
appendcB_in_gaa(x1, x2, x3)  =  appendcB_in_gaa(x1)
[]  =  []
appendcB_out_gaa(x1, x2, x3)  =  appendcB_out_gaa(x1, x3)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
appendcD_out_agaa(x1, x2, x3, x4)  =  appendcD_out_agaa(x2, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
appendcE_in_aa(x1, x2)  =  appendcE_in_aa
appendcE_out_aa(x1, x2)  =  appendcE_out_aa(x2)
GOALF_IN_GAA(x1, x2, x3)  =  GOALF_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
APPENDB_IN_GAA(x1, x2, x3)  =  APPENDB_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
LASTC_IN_AG(x1, x2)  =  LASTC_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U9_GAA(x1, x2, x3)  =  U9_GAA(x3)
U10_GAA(x1, x2, x3)  =  U10_GAA(x3)

We have to consider all (P,R,Pi)-chains

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LASTC_IN_AG(X1, .(X2, X3)) → LASTC_IN_AG(X1, X3)

The TRS R consists of the following rules:

s2lcA_in_ga(s(X1), .(X2, X3)) → U12_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U12_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
appendcD_in_agaa(X1, X2, X3, .(X1, X4)) → U15_agaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
appendcB_in_gaa([], X1, .(X1, [])) → appendcB_out_gaa([], X1, .(X1, []))
appendcB_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcB_out_gaa(.(X1, X2), X3, .(X1, X4))
U15_agaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcD_out_agaa(X1, X2, X3, .(X1, X4))
appendcE_in_aa(X1, .(X1, [])) → appendcE_out_aa(X1, .(X1, []))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
appendcD_in_agaa(x1, x2, x3, x4)  =  appendcD_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5)  =  U15_agaa(x2, x5)
appendcB_in_gaa(x1, x2, x3)  =  appendcB_in_gaa(x1)
[]  =  []
appendcB_out_gaa(x1, x2, x3)  =  appendcB_out_gaa(x1, x3)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
appendcD_out_agaa(x1, x2, x3, x4)  =  appendcD_out_agaa(x2, x4)
appendcE_in_aa(x1, x2)  =  appendcE_in_aa
appendcE_out_aa(x1, x2)  =  appendcE_out_aa(x2)
LASTC_IN_AG(x1, x2)  =  LASTC_IN_AG(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:

LASTC_IN_AG(X1, .(X2, X3)) → LASTC_IN_AG(X1, X3)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LASTC_IN_AG(x1, x2)  =  LASTC_IN_AG(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:

LASTC_IN_AG(.(X3)) → LASTC_IN_AG(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:

  • LASTC_IN_AG(.(X3)) → LASTC_IN_AG(X3)
    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_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDB_IN_GAA(X2, X3, X4)

The TRS R consists of the following rules:

s2lcA_in_ga(s(X1), .(X2, X3)) → U12_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U12_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
appendcD_in_agaa(X1, X2, X3, .(X1, X4)) → U15_agaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
appendcB_in_gaa([], X1, .(X1, [])) → appendcB_out_gaa([], X1, .(X1, []))
appendcB_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcB_out_gaa(.(X1, X2), X3, .(X1, X4))
U15_agaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcD_out_agaa(X1, X2, X3, .(X1, X4))
appendcE_in_aa(X1, .(X1, [])) → appendcE_out_aa(X1, .(X1, []))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
appendcD_in_agaa(x1, x2, x3, x4)  =  appendcD_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5)  =  U15_agaa(x2, x5)
appendcB_in_gaa(x1, x2, x3)  =  appendcB_in_gaa(x1)
[]  =  []
appendcB_out_gaa(x1, x2, x3)  =  appendcB_out_gaa(x1, x3)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
appendcD_out_agaa(x1, x2, x3, x4)  =  appendcD_out_agaa(x2, x4)
appendcE_in_aa(x1, x2)  =  appendcE_in_aa
appendcE_out_aa(x1, x2)  =  appendcE_out_aa(x2)
APPENDB_IN_GAA(x1, x2, x3)  =  APPENDB_IN_GAA(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:

APPENDB_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDB_IN_GAA(X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPENDB_IN_GAA(x1, x2, x3)  =  APPENDB_IN_GAA(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:

APPENDB_IN_GAA(.(X2)) → APPENDB_IN_GAA(X2)

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_GAA(.(X2)) → APPENDB_IN_GAA(X2)
    The graph contains the following edges 1 > 1

(20) YES

(21) 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)) → U12_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
U12_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))
appendcD_in_agaa(X1, X2, X3, .(X1, X4)) → U15_agaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
appendcB_in_gaa([], X1, .(X1, [])) → appendcB_out_gaa([], X1, .(X1, []))
appendcB_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcB_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcB_out_gaa(.(X1, X2), X3, .(X1, X4))
U15_agaa(X1, X2, X3, X4, appendcB_out_gaa(X2, X3, X4)) → appendcD_out_agaa(X1, X2, X3, .(X1, X4))
appendcE_in_aa(X1, .(X1, [])) → appendcE_out_aa(X1, .(X1, []))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
appendcD_in_agaa(x1, x2, x3, x4)  =  appendcD_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5)  =  U15_agaa(x2, x5)
appendcB_in_gaa(x1, x2, x3)  =  appendcB_in_gaa(x1)
[]  =  []
appendcB_out_gaa(x1, x2, x3)  =  appendcB_out_gaa(x1, x3)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
appendcD_out_agaa(x1, x2, x3, x4)  =  appendcD_out_agaa(x2, x4)
appendcE_in_aa(x1, x2)  =  appendcE_in_aa
appendcE_out_aa(x1, x2)  =  appendcE_out_aa(x2)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)

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:

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

(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:

S2LA_IN_GA(s(X1)) → S2LA_IN_GA(X1)

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:

  • S2LA_IN_GA(s(X1)) → S2LA_IN_GA(X1)
    The graph contains the following edges 1 > 1

(27) YES