Left Termination proof of /tmp/tmpEJ5xGB/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)

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)

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)

↳ 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)

↳ 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:

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

↳ 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)

↳ 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)

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

↳ 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)

↳ 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)

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