Left Termination proof of /tmp/tmpCUom7N/applast.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

goal₃(A, B, C) :- s2l₂(A, D), applast₃(D, B, C).
applast₃(L, X, Last) :- append₃(L, .₂(X, {}₀), LX), last₂(Last, LX).
last₂(X, .₂(X, {}₀)).
last₂(X, .₂(H, T)) :- last₂(X, T).
append₃({}₀, L, L).
append₃(.₂(H, L1), L2, .₂(H, L3)) :- append₃(L1, L2, L3).
s2l₂(s₁(X), .₂(Y, Xs)) :- s2l₂(X, Xs).
s2l₂(0₀, {}₀).

With regard to the inferred argument filtering the predicates were used in the following modes:
goal₃: (b,f,f)
s2l₂: (b,f)
applast₃: (b,f,f)
append₃: (b,b,f)
last₂: (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_3_in_gaa₃(A, B, C) -> if_goal_3_in_1_gaa₄(A, B, C, s2l_2_in_ga₂(A, D))
s2l_2_in_ga₂(s_1₁(X), ._2₂(Y, Xs)) -> if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
s2l_2_in_ga₂(0_0, []_0) -> s2l_2_out_ga₂(0_0, []_0)
if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_out_ga₂(X, Xs)) -> s2l_2_out_ga₂(s_1₁(X), ._2₂(Y, Xs))
if_goal_3_in_1_gaa₄(A, B, C, s2l_2_out_ga₂(A, D)) -> if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
applast_3_in_gaa₃(L, X, Last) -> if_applast_3_in_1_gaa₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_out_gga₃(L1, L2, L3)) -> append_3_out_gga₃(._2₂(H, L1), L2, ._2₂(H, L3))
if_applast_3_in_1_gaa₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
last_2_in_ag₂(X, ._2₂(X, []_0)) -> last_2_out_ag₂(X, ._2₂(X, []_0))
last_2_in_ag₂(X, ._2₂(H, T)) -> if_last_2_in_1_ag₄(X, H, T, last_2_in_ag₂(X, T))
if_last_2_in_1_ag₄(X, H, T, last_2_out_ag₂(X, T)) -> last_2_out_ag₂(X, ._2₂(H, T))
if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_out_ag₂(Last, LX)) -> applast_3_out_gaa₃(L, X, Last)
if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_out_gaa₃(D, B, C)) -> goal_3_out_gaa₃(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_3_in_gaa₃(A, B, C) -> if_goal_3_in_1_gaa₄(A, B, C, s2l_2_in_ga₂(A, D))
s2l_2_in_ga₂(s_1₁(X), ._2₂(Y, Xs)) -> if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
s2l_2_in_ga₂(0_0, []_0) -> s2l_2_out_ga₂(0_0, []_0)
if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_out_ga₂(X, Xs)) -> s2l_2_out_ga₂(s_1₁(X), ._2₂(Y, Xs))
if_goal_3_in_1_gaa₄(A, B, C, s2l_2_out_ga₂(A, D)) -> if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
applast_3_in_gaa₃(L, X, Last) -> if_applast_3_in_1_gaa₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_out_gga₃(L1, L2, L3)) -> append_3_out_gga₃(._2₂(H, L1), L2, ._2₂(H, L3))
if_applast_3_in_1_gaa₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
last_2_in_ag₂(X, ._2₂(X, []_0)) -> last_2_out_ag₂(X, ._2₂(X, []_0))
last_2_in_ag₂(X, ._2₂(H, T)) -> if_last_2_in_1_ag₄(X, H, T, last_2_in_ag₂(X, T))
if_last_2_in_1_ag₄(X, H, T, last_2_out_ag₂(X, T)) -> last_2_out_ag₂(X, ._2₂(H, T))
if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_out_ag₂(Last, LX)) -> applast_3_out_gaa₃(L, X, Last)
if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_out_gaa₃(D, B, C)) -> goal_3_out_gaa₃(A, B, C)

GOAL_3_IN_GAA₃(A, B, C) -> IF_GOAL_3_IN_1_GAA₄(A, B, C, s2l_2_in_ga₂(A, D))
GOAL_3_IN_GAA₃(A, B, C) -> S2L_2_IN_GA₂(A, D)
S2L_2_IN_GA₂(s_1₁(X), ._2₂(Y, Xs)) -> IF_S2L_2_IN_1_GA₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
S2L_2_IN_GA₂(s_1₁(X), ._2₂(Y, Xs)) -> S2L_2_IN_GA₂(X, Xs)
IF_GOAL_3_IN_1_GAA₄(A, B, C, s2l_2_out_ga₂(A, D)) -> IF_GOAL_3_IN_2_GAA₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
IF_GOAL_3_IN_1_GAA₄(A, B, C, s2l_2_out_ga₂(A, D)) -> APPLAST_3_IN_GAA₃(D, B, C)
APPLAST_3_IN_GAA₃(L, X, Last) -> IF_APPLAST_3_IN_1_GAA₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
APPLAST_3_IN_GAA₃(L, X, Last) -> APPEND_3_IN_GGA₃(L, ._2₂(X, []_0), LX)
APPEND_3_IN_GGA₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> IF_APPEND_3_IN_1_GGA₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
APPEND_3_IN_GGA₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> APPEND_3_IN_GGA₃(L1, L2, L3)
IF_APPLAST_3_IN_1_GAA₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> IF_APPLAST_3_IN_2_GAA₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
IF_APPLAST_3_IN_1_GAA₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> LAST_2_IN_AG₂(Last, LX)
LAST_2_IN_AG₂(X, ._2₂(H, T)) -> IF_LAST_2_IN_1_AG₄(X, H, T, last_2_in_ag₂(X, T))
LAST_2_IN_AG₂(X, ._2₂(H, T)) -> LAST_2_IN_AG₂(X, T)

The TRS R consists of the following rules:

goal_3_in_gaa₃(A, B, C) -> if_goal_3_in_1_gaa₄(A, B, C, s2l_2_in_ga₂(A, D))
s2l_2_in_ga₂(s_1₁(X), ._2₂(Y, Xs)) -> if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
s2l_2_in_ga₂(0_0, []_0) -> s2l_2_out_ga₂(0_0, []_0)
if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_out_ga₂(X, Xs)) -> s2l_2_out_ga₂(s_1₁(X), ._2₂(Y, Xs))
if_goal_3_in_1_gaa₄(A, B, C, s2l_2_out_ga₂(A, D)) -> if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
applast_3_in_gaa₃(L, X, Last) -> if_applast_3_in_1_gaa₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_out_gga₃(L1, L2, L3)) -> append_3_out_gga₃(._2₂(H, L1), L2, ._2₂(H, L3))
if_applast_3_in_1_gaa₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
last_2_in_ag₂(X, ._2₂(X, []_0)) -> last_2_out_ag₂(X, ._2₂(X, []_0))
last_2_in_ag₂(X, ._2₂(H, T)) -> if_last_2_in_1_ag₄(X, H, T, last_2_in_ag₂(X, T))
if_last_2_in_1_ag₄(X, H, T, last_2_out_ag₂(X, T)) -> last_2_out_ag₂(X, ._2₂(H, T))
if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_out_ag₂(Last, LX)) -> applast_3_out_gaa₃(L, X, Last)
if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_out_gaa₃(D, B, C)) -> goal_3_out_gaa₃(A, B, C)

The argument filtering Pi contains the following mapping:
goal_3_in_gaa₃(x1, x2, x3) = goal_3_in_gaa₁(x1)
._2₂(x1, x2) = ._2₁(x2)
[]_0 = []_0
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
if_goal_3_in_1_gaa₄(x1, x2, x3, x4) = if_goal_3_in_1_gaa₁(x4)
s2l_2_in_ga₂(x1, x2) = s2l_2_in_ga₁(x1)
if_s2l_2_in_1_ga₄(x1, x2, x3, x4) = if_s2l_2_in_1_ga₁(x4)
s2l_2_out_ga₂(x1, x2) = s2l_2_out_ga₁(x2)
if_goal_3_in_2_gaa₅(x1, x2, x3, x4, x5) = if_goal_3_in_2_gaa₁(x5)
applast_3_in_gaa₃(x1, x2, x3) = applast_3_in_gaa₁(x1)
if_applast_3_in_1_gaa₄(x1, x2, x3, x4) = if_applast_3_in_1_gaa₁(x4)
append_3_in_gga₃(x1, x2, x3) = append_3_in_gga₂(x1, x2)
append_3_out_gga₃(x1, x2, x3) = append_3_out_gga₁(x3)
if_append_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_append_3_in_1_gga₁(x5)
if_applast_3_in_2_gaa₅(x1, x2, x3, x4, x5) = if_applast_3_in_2_gaa₁(x5)
last_2_in_ag₂(x1, x2) = last_2_in_ag₁(x2)
last_2_out_ag₂(x1, x2) = last_2_out_ag
if_last_2_in_1_ag₄(x1, x2, x3, x4) = if_last_2_in_1_ag₁(x4)
applast_3_out_gaa₃(x1, x2, x3) = applast_3_out_gaa
goal_3_out_gaa₃(x1, x2, x3) = goal_3_out_gaa
S2L_2_IN_GA₂(x1, x2) = S2L_2_IN_GA₁(x1)
IF_APPLAST_3_IN_1_GAA₄(x1, x2, x3, x4) = IF_APPLAST_3_IN_1_GAA₁(x4)
IF_GOAL_3_IN_1_GAA₄(x1, x2, x3, x4) = IF_GOAL_3_IN_1_GAA₁(x4)
IF_APPEND_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_APPEND_3_IN_1_GGA₁(x5)
IF_APPLAST_3_IN_2_GAA₅(x1, x2, x3, x4, x5) = IF_APPLAST_3_IN_2_GAA₁(x5)
APPLAST_3_IN_GAA₃(x1, x2, x3) = APPLAST_3_IN_GAA₁(x1)
IF_GOAL_3_IN_2_GAA₅(x1, x2, x3, x4, x5) = IF_GOAL_3_IN_2_GAA₁(x5)
GOAL_3_IN_GAA₃(x1, x2, x3) = GOAL_3_IN_GAA₁(x1)
IF_S2L_2_IN_1_GA₄(x1, x2, x3, x4) = IF_S2L_2_IN_1_GA₁(x4)
IF_LAST_2_IN_1_AG₄(x1, x2, x3, x4) = IF_LAST_2_IN_1_AG₁(x4)
LAST_2_IN_AG₂(x1, x2) = LAST_2_IN_AG₁(x2)
APPEND_3_IN_GGA₃(x1, x2, x3) = APPEND_3_IN_GGA₂(x1, x2)

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_3_IN_GAA₃(A, B, C) -> IF_GOAL_3_IN_1_GAA₄(A, B, C, s2l_2_in_ga₂(A, D))
GOAL_3_IN_GAA₃(A, B, C) -> S2L_2_IN_GA₂(A, D)
S2L_2_IN_GA₂(s_1₁(X), ._2₂(Y, Xs)) -> IF_S2L_2_IN_1_GA₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
S2L_2_IN_GA₂(s_1₁(X), ._2₂(Y, Xs)) -> S2L_2_IN_GA₂(X, Xs)
IF_GOAL_3_IN_1_GAA₄(A, B, C, s2l_2_out_ga₂(A, D)) -> IF_GOAL_3_IN_2_GAA₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
IF_GOAL_3_IN_1_GAA₄(A, B, C, s2l_2_out_ga₂(A, D)) -> APPLAST_3_IN_GAA₃(D, B, C)
APPLAST_3_IN_GAA₃(L, X, Last) -> IF_APPLAST_3_IN_1_GAA₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
APPLAST_3_IN_GAA₃(L, X, Last) -> APPEND_3_IN_GGA₃(L, ._2₂(X, []_0), LX)
APPEND_3_IN_GGA₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> IF_APPEND_3_IN_1_GGA₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
APPEND_3_IN_GGA₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> APPEND_3_IN_GGA₃(L1, L2, L3)
IF_APPLAST_3_IN_1_GAA₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> IF_APPLAST_3_IN_2_GAA₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
IF_APPLAST_3_IN_1_GAA₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> LAST_2_IN_AG₂(Last, LX)
LAST_2_IN_AG₂(X, ._2₂(H, T)) -> IF_LAST_2_IN_1_AG₄(X, H, T, last_2_in_ag₂(X, T))
LAST_2_IN_AG₂(X, ._2₂(H, T)) -> LAST_2_IN_AG₂(X, T)

The TRS R consists of the following rules:

goal_3_in_gaa₃(A, B, C) -> if_goal_3_in_1_gaa₄(A, B, C, s2l_2_in_ga₂(A, D))
s2l_2_in_ga₂(s_1₁(X), ._2₂(Y, Xs)) -> if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
s2l_2_in_ga₂(0_0, []_0) -> s2l_2_out_ga₂(0_0, []_0)
if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_out_ga₂(X, Xs)) -> s2l_2_out_ga₂(s_1₁(X), ._2₂(Y, Xs))
if_goal_3_in_1_gaa₄(A, B, C, s2l_2_out_ga₂(A, D)) -> if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
applast_3_in_gaa₃(L, X, Last) -> if_applast_3_in_1_gaa₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_out_gga₃(L1, L2, L3)) -> append_3_out_gga₃(._2₂(H, L1), L2, ._2₂(H, L3))
if_applast_3_in_1_gaa₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
last_2_in_ag₂(X, ._2₂(X, []_0)) -> last_2_out_ag₂(X, ._2₂(X, []_0))
last_2_in_ag₂(X, ._2₂(H, T)) -> if_last_2_in_1_ag₄(X, H, T, last_2_in_ag₂(X, T))
if_last_2_in_1_ag₄(X, H, T, last_2_out_ag₂(X, T)) -> last_2_out_ag₂(X, ._2₂(H, T))
if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_out_ag₂(Last, LX)) -> applast_3_out_gaa₃(L, X, Last)
if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_out_gaa₃(D, B, C)) -> goal_3_out_gaa₃(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:

LAST_2_IN_AG₂(X, ._2₂(H, T)) -> LAST_2_IN_AG₂(X, T)

The TRS R consists of the following rules:

goal_3_in_gaa₃(A, B, C) -> if_goal_3_in_1_gaa₄(A, B, C, s2l_2_in_ga₂(A, D))
s2l_2_in_ga₂(s_1₁(X), ._2₂(Y, Xs)) -> if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
s2l_2_in_ga₂(0_0, []_0) -> s2l_2_out_ga₂(0_0, []_0)
if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_out_ga₂(X, Xs)) -> s2l_2_out_ga₂(s_1₁(X), ._2₂(Y, Xs))
if_goal_3_in_1_gaa₄(A, B, C, s2l_2_out_ga₂(A, D)) -> if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
applast_3_in_gaa₃(L, X, Last) -> if_applast_3_in_1_gaa₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_out_gga₃(L1, L2, L3)) -> append_3_out_gga₃(._2₂(H, L1), L2, ._2₂(H, L3))
if_applast_3_in_1_gaa₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
last_2_in_ag₂(X, ._2₂(X, []_0)) -> last_2_out_ag₂(X, ._2₂(X, []_0))
last_2_in_ag₂(X, ._2₂(H, T)) -> if_last_2_in_1_ag₄(X, H, T, last_2_in_ag₂(X, T))
if_last_2_in_1_ag₄(X, H, T, last_2_out_ag₂(X, T)) -> last_2_out_ag₂(X, ._2₂(H, T))
if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_out_ag₂(Last, LX)) -> applast_3_out_gaa₃(L, X, Last)
if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_out_gaa₃(D, B, C)) -> goal_3_out_gaa₃(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:

LAST_2_IN_AG₂(X, ._2₂(H, T)) -> LAST_2_IN_AG₂(X, T)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₁(x2)
LAST_2_IN_AG₂(x1, x2) = LAST_2_IN_AG₁(x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem 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:

LAST_2_IN_AG₁(._2₁(T)) -> LAST_2_IN_AG₁(T)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LAST_2_IN_AG₁}.
By using the subterm criterion together with the size-change analysis 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:

LAST_2_IN_AG₁(._2₁(T)) -> LAST_2_IN_AG₁(T)
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:

APPEND_3_IN_GGA₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> APPEND_3_IN_GGA₃(L1, L2, L3)

The TRS R consists of the following rules:

goal_3_in_gaa₃(A, B, C) -> if_goal_3_in_1_gaa₄(A, B, C, s2l_2_in_ga₂(A, D))
s2l_2_in_ga₂(s_1₁(X), ._2₂(Y, Xs)) -> if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
s2l_2_in_ga₂(0_0, []_0) -> s2l_2_out_ga₂(0_0, []_0)
if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_out_ga₂(X, Xs)) -> s2l_2_out_ga₂(s_1₁(X), ._2₂(Y, Xs))
if_goal_3_in_1_gaa₄(A, B, C, s2l_2_out_ga₂(A, D)) -> if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
applast_3_in_gaa₃(L, X, Last) -> if_applast_3_in_1_gaa₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_out_gga₃(L1, L2, L3)) -> append_3_out_gga₃(._2₂(H, L1), L2, ._2₂(H, L3))
if_applast_3_in_1_gaa₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
last_2_in_ag₂(X, ._2₂(X, []_0)) -> last_2_out_ag₂(X, ._2₂(X, []_0))
last_2_in_ag₂(X, ._2₂(H, T)) -> if_last_2_in_1_ag₄(X, H, T, last_2_in_ag₂(X, T))
if_last_2_in_1_ag₄(X, H, T, last_2_out_ag₂(X, T)) -> last_2_out_ag₂(X, ._2₂(H, T))
if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_out_ag₂(Last, LX)) -> applast_3_out_gaa₃(L, X, Last)
if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_out_gaa₃(D, B, C)) -> goal_3_out_gaa₃(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:

APPEND_3_IN_GGA₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> APPEND_3_IN_GGA₃(L1, L2, L3)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₁(x2)
APPEND_3_IN_GGA₃(x1, x2, x3) = APPEND_3_IN_GGA₂(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem 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:

APPEND_3_IN_GGA₂(._2₁(L1), L2) -> APPEND_3_IN_GGA₂(L1, L2)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APPEND_3_IN_GGA₂}.
By using the subterm criterion together with the size-change analysis 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:

APPEND_3_IN_GGA₂(._2₁(L1), L2) -> APPEND_3_IN_GGA₂(L1, L2)
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:

S2L_2_IN_GA₂(s_1₁(X), ._2₂(Y, Xs)) -> S2L_2_IN_GA₂(X, Xs)

The TRS R consists of the following rules:

goal_3_in_gaa₃(A, B, C) -> if_goal_3_in_1_gaa₄(A, B, C, s2l_2_in_ga₂(A, D))
s2l_2_in_ga₂(s_1₁(X), ._2₂(Y, Xs)) -> if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_in_ga₂(X, Xs))
s2l_2_in_ga₂(0_0, []_0) -> s2l_2_out_ga₂(0_0, []_0)
if_s2l_2_in_1_ga₄(X, Y, Xs, s2l_2_out_ga₂(X, Xs)) -> s2l_2_out_ga₂(s_1₁(X), ._2₂(Y, Xs))
if_goal_3_in_1_gaa₄(A, B, C, s2l_2_out_ga₂(A, D)) -> if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_in_gaa₃(D, B, C))
applast_3_in_gaa₃(L, X, Last) -> if_applast_3_in_1_gaa₄(L, X, Last, append_3_in_gga₃(L, ._2₂(X, []_0), LX))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L1), L2, ._2₂(H, L3)) -> if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_in_gga₃(L1, L2, L3))
if_append_3_in_1_gga₅(H, L1, L2, L3, append_3_out_gga₃(L1, L2, L3)) -> append_3_out_gga₃(._2₂(H, L1), L2, ._2₂(H, L3))
if_applast_3_in_1_gaa₄(L, X, Last, append_3_out_gga₃(L, ._2₂(X, []_0), LX)) -> if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_in_ag₂(Last, LX))
last_2_in_ag₂(X, ._2₂(X, []_0)) -> last_2_out_ag₂(X, ._2₂(X, []_0))
last_2_in_ag₂(X, ._2₂(H, T)) -> if_last_2_in_1_ag₄(X, H, T, last_2_in_ag₂(X, T))
if_last_2_in_1_ag₄(X, H, T, last_2_out_ag₂(X, T)) -> last_2_out_ag₂(X, ._2₂(H, T))
if_applast_3_in_2_gaa₅(L, X, Last, LX, last_2_out_ag₂(Last, LX)) -> applast_3_out_gaa₃(L, X, Last)
if_goal_3_in_2_gaa₅(A, B, C, D, applast_3_out_gaa₃(D, B, C)) -> goal_3_out_gaa₃(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:

S2L_2_IN_GA₂(s_1₁(X), ._2₂(Y, Xs)) -> S2L_2_IN_GA₂(X, Xs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₁(x2)
s_1₁(x1) = s_1₁(x1)
S2L_2_IN_GA₂(x1, x2) = S2L_2_IN_GA₁(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem 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:

S2L_2_IN_GA₁(s_1₁(X)) -> S2L_2_IN_GA₁(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {S2L_2_IN_GA₁}.
By using the subterm criterion together with the size-change analysis 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:

S2L_2_IN_GA₁(s_1₁(X)) -> S2L_2_IN_GA₁(X)
The graph contains the following edges 1 > 1