YES Left Termination proof of ../tpdb/LP/SGST06/btapplast.pl
Left Termination of the query pattern goal_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

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).

We use the technique of [30]. 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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


Using Dependency Pairs [1,30] 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
TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x3, x6)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x5)
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
TAPPLAST_IN_GAA(x1, x2, x3)  =  TAPPLAST_IN_GAA(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x6)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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
TAPPEND_IN_GGA(x1, x2, x3)  =  TAPPEND_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x3, x6)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x5)
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
TAPPLAST_IN_GAA(x1, x2, x3)  =  TAPPLAST_IN_GAA(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 15 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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

TLAST_IN_AG(X, node(L, H, R)) → TLAST_IN_AG(X, L)
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
TLAST_IN_AG(x1, x2)  =  TLAST_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

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

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

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

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

S2T_IN_GA(s(X), node(T, Y, nil)) → 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)) → 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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

S2T_IN_GA(s(X), node(T, Y, nil)) → 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)) → 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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: