Left Termination of the query pattern goal_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
      ↳10 PiDP
        ↳11 UsableRulesProof (⇔)
        ↳12 PiDP
      ↳13 PiDP
        ↳14 UsableRulesProof (⇔)
        ↳15 PiDP
  ↳16 PrologToPiTRSProof (⇐)
    ↳17 PiTRS
    ↳18 DependencyPairsProof (⇔)
    ↳19 PiDP
    ↳20 DependencyGraphProof (⇔)
    ↳21 AND
      ↳22 PiDP
        ↳23 UsableRulesProof (⇔)
        ↳24 PiDP
        ↳25 PiDPToQDPProof (⇐)
        ↳26 QDP
        ↳27 QDPSizeChangeProof (⇔)
        ↳28 TRUE
      ↳29 PiDP
        ↳30 UsableRulesProof (⇔)
        ↳31 PiDP
        ↳32 PiDPToQDPProof (⇐)
        ↳33 QDP
        ↳34 QDPSizeChangeProof (⇔)
        ↳35 TRUE
      ↳36 PiDP
        ↳37 UsableRulesProof (⇔)
        ↳38 PiDP
        ↳39 PiDPToQDPProof (⇐)
        ↳40 QDP
        ↳41 QDPSizeChangeProof (⇔)
        ↳42 TRUE

(0) Obligation:

Clauses:

goal(A, B, C) :- ','(s2t(A, T), tapplast(T, B, C)).
tapplast(L, X, Last) :- ','(tappend(L, node(nil, X, nil), LX), tlast(Last, LX)).
tlast(X, node(nil, X, nil)).
tlast(X, node(L, H, R)) :- tlast(X, L).
tlast(X, node(L, H, R)) :- tlast(X, R).
tappend(nil, T, T).
tappend(node(nil, X, T2), T1, node(T1, X, T2)).
tappend(node(T1, X, nil), T2, node(T1, X, T2)).
tappend(node(T1, X, T2), T3, node(U, X, T2)) :- tappend(T1, T3, U).
tappend(node(T1, X, T2), T3, node(T1, X, U)) :- tappend(T2, T3, U).
s2t(s(X), node(T, Y, T)) :- s2t(X, T).
s2t(s(X), node(nil, Y, T)) :- s2t(X, T).
s2t(s(X), node(T, Y, nil)) :- s2t(X, T).
s2t(s(X), node(nil, Y, nil)).
s2t(0, nil).

Queries:

goal(g,a,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b,f,f)
s2t_in: (b,f)
tapplast_in: (b,f,f)
tappend_in: (b,b,f)
tlast_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x1, x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x3, x4, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x3, x4, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag(x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x2, x4, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x2, x4, x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa(x1)
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x1, x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x3, x4, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x3, x4, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag(x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x2, x4, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x2, x4, x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa(x1)
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U9_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U10_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U11_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_gaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_GAA(T, B, C)
TAPPLAST_IN_GAA(L, X, Last) → U3_GAA(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
TAPPLAST_IN_GAA(L, X, Last) → TAPPEND_IN_GGA(L, node(nil, X, nil), LX)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → U7_GGA(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_GGA(T1, T3, U)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → U8_GGA(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_GGA(T2, T3, U)
U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_GAA(L, X, Last, tlast_in_ag(Last, LX))
U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → TLAST_IN_AG(Last, LX)
TLAST_IN_AG(X, node(L, H, R)) → U5_AG(X, L, H, R, tlast_in_ag(X, L))
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)
TLAST_IN_AG(X, node(L, H, R)) → U6_AG(X, L, H, R, tlast_in_ag(X, R))
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, R)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x1, x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x3, x4, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x3, x4, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag(x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x2, x4, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x2, x4, x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa(x1)
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x1, x4)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x1, x4)
TAPPLAST_IN_GAA(x1, x2, x3)  =  TAPPLAST_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x1, x4)
TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x1, x3, x4, x6)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x3, x4, x6)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x2, x4, x5)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x2, x4, x5)

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

(4) Obligation:

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

GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U9_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U10_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U11_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_gaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_GAA(T, B, C)
TAPPLAST_IN_GAA(L, X, Last) → U3_GAA(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
TAPPLAST_IN_GAA(L, X, Last) → TAPPEND_IN_GGA(L, node(nil, X, nil), LX)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → U7_GGA(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_GGA(T1, T3, U)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → U8_GGA(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_GGA(T2, T3, U)
U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_GAA(L, X, Last, tlast_in_ag(Last, LX))
U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → TLAST_IN_AG(Last, LX)
TLAST_IN_AG(X, node(L, H, R)) → U5_AG(X, L, H, R, tlast_in_ag(X, L))
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)
TLAST_IN_AG(X, node(L, H, R)) → U6_AG(X, L, H, R, tlast_in_ag(X, R))
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, R)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x1, x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x3, x4, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x3, x4, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag(x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x2, x4, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x2, x4, x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa(x1)
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x1, x4)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x1, x4)
TAPPLAST_IN_GAA(x1, x2, x3)  =  TAPPLAST_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x1, x4)
TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x1, x3, x4, x6)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x3, x4, x6)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x2, x4, x5)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x2, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, R)
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x1, x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x3, x4, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x3, x4, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag(x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x2, x4, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x2, x4, x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa(x1)
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
TLAST_IN_AG(x1, x2)  =  TLAST_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:

TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, R)
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)

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

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

(10) Obligation:

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

TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_GGA(T2, T3, U)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_GGA(T1, T3, U)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x1, x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x3, x4, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x3, x4, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag(x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x2, x4, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x2, x4, x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa(x1)
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)

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

(11) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(12) Obligation:

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

TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_GGA(T2, T3, U)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_GGA(T1, T3, U)

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

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

(13) Obligation:

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

S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x1, x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x3, x4, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x3, x4, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag(x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x2, x4, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x2, x4, x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa(x1)
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)

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

(14) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(15) Obligation:

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

S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)

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

(16) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b,f,f)
s2t_in: (b,f)
tapplast_in: (b,f,f)
tappend_in: (b,b,f)
tlast_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(17) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa

(18) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U9_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U10_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U11_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_gaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_GAA(T, B, C)
TAPPLAST_IN_GAA(L, X, Last) → U3_GAA(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
TAPPLAST_IN_GAA(L, X, Last) → TAPPEND_IN_GGA(L, node(nil, X, nil), LX)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → U7_GGA(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_GGA(T1, T3, U)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → U8_GGA(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_GGA(T2, T3, U)
U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_GAA(L, X, Last, tlast_in_ag(Last, LX))
U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → TLAST_IN_AG(Last, LX)
TLAST_IN_AG(X, node(L, H, R)) → U5_AG(X, L, H, R, tlast_in_ag(X, L))
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)
TLAST_IN_AG(X, node(L, H, R)) → U6_AG(X, L, H, R, tlast_in_ag(X, R))
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, R)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
TAPPLAST_IN_GAA(x1, x2, x3)  =  TAPPLAST_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x3, x6)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x6)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x4)
TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x5)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)

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

(19) Obligation:

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

GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U9_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U10_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U11_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_gaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_GAA(T, B, C)
TAPPLAST_IN_GAA(L, X, Last) → U3_GAA(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
TAPPLAST_IN_GAA(L, X, Last) → TAPPEND_IN_GGA(L, node(nil, X, nil), LX)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → U7_GGA(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_GGA(T1, T3, U)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → U8_GGA(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_GGA(T2, T3, U)
U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_GAA(L, X, Last, tlast_in_ag(Last, LX))
U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → TLAST_IN_AG(Last, LX)
TLAST_IN_AG(X, node(L, H, R)) → U5_AG(X, L, H, R, tlast_in_ag(X, L))
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)
TLAST_IN_AG(X, node(L, H, R)) → U6_AG(X, L, H, R, tlast_in_ag(X, R))
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, R)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
TAPPLAST_IN_GAA(x1, x2, x3)  =  TAPPLAST_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x3, x6)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x6)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x4)
TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x5)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)

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

(20) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.

(21) Complex Obligation (AND)

(22) Obligation:

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

TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, R)
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)

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

(23) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(24) Obligation:

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

TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, R)
TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)

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

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

(25) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(26) Obligation:

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

TLAST_IN_AG(node(L, R)) → TLAST_IN_AG(R)
TLAST_IN_AG(node(L, R)) → TLAST_IN_AG(L)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(27) 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:

  • TLAST_IN_AG(node(L, R)) → TLAST_IN_AG(R)
    The graph contains the following edges 1 > 1

  • TLAST_IN_AG(node(L, R)) → TLAST_IN_AG(L)
    The graph contains the following edges 1 > 1

(28) TRUE

(29) Obligation:

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

TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_GGA(T2, T3, U)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_GGA(T1, T3, U)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)

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

(30) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(31) Obligation:

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

TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_GGA(T2, T3, U)
TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_GGA(T1, T3, U)

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

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

(32) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(33) Obligation:

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

TAPPEND_IN_GGA(node(T1, T2), T3) → TAPPEND_IN_GGA(T2, T3)
TAPPEND_IN_GGA(node(T1, T2), T3) → TAPPEND_IN_GGA(T1, T3)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(34) 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:

  • TAPPEND_IN_GGA(node(T1, T2), T3) → TAPPEND_IN_GGA(T2, T3)
    The graph contains the following edges 1 > 1, 2 >= 2

  • TAPPEND_IN_GGA(node(T1, T2), T3) → TAPPEND_IN_GGA(T1, T3)
    The graph contains the following edges 1 > 1, 2 >= 2

(35) TRUE

(36) Obligation:

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

S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_gaa(T, B, C))
tapplast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX))
tappend_in_gga(nil, T, T) → tappend_out_gga(nil, T, T)
tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) → U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U))
tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) → U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U))
U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U))
U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) → tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2))
U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) → U4_gaa(L, X, Last, tlast_in_ag(Last, LX))
tlast_in_ag(X, node(nil, X, nil)) → tlast_out_ag(X, node(nil, X, nil))
tlast_in_ag(X, node(L, H, R)) → U5_ag(X, L, H, R, tlast_in_ag(X, L))
tlast_in_ag(X, node(L, H, R)) → U6_ag(X, L, H, R, tlast_in_ag(X, R))
U6_ag(X, L, H, R, tlast_out_ag(X, R)) → tlast_out_ag(X, node(L, H, R))
U5_ag(X, L, H, R, tlast_out_ag(X, L)) → tlast_out_ag(X, node(L, H, R))
U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) → tapplast_out_gaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_gaa(x1, x2, x3)  =  tapplast_in_gaa(x1)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
tappend_in_gga(x1, x2, x3)  =  tappend_in_gga(x1, x2)
nil  =  nil
tappend_out_gga(x1, x2, x3)  =  tappend_out_gga(x3)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x3, x6)
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x1, x6)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
tlast_in_ag(x1, x2)  =  tlast_in_ag(x2)
tlast_out_ag(x1, x2)  =  tlast_out_ag
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
tapplast_out_gaa(x1, x2, x3)  =  tapplast_out_gaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)

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

(37) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(38) Obligation:

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

S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)

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

(39) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(40) Obligation:

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

S2T_IN_GA(s(X)) → S2T_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(41) 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:

  • S2T_IN_GA(s(X)) → S2T_IN_GA(X)
    The graph contains the following edges 1 > 1

(42) TRUE