Left Termination proof of /tmp/tmp1MTOF

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

↳ PROLOG

  ↳ PrologToPiTRSProof

shanoi₅(s₁(0₀), A, B, C, .₂(mv₂(A, C), {}₀)).
shanoi₅(s₁²(X), A, B, C, M) :- eq₂(N1, s₁(X)), shanoi₅(N1, A, C, B, M1), shanoi₅(N1, B, A, C, M2), append₃(M1, .₂(mv₂(A, C), {}₀), T), append₃(T, M2, M).
append₃({}₀, L, L).
append₃(.₂(H, L), L1, .₂(H, R)) :- append₃(L, L1, R).
eq₂(X, X).

With regard to the inferred argument filtering the predicates were used in the following modes:
shanoi₅: (b,b,b,b,f)
append₃: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

shanoi_5_in_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0)) -> shanoi_5_out_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0))
shanoi_5_in_gggga₅(s_1₁(s_1₁(X)), A, B, C, M) -> if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
eq_2_in_ag₂(X, X) -> eq_2_out_ag₂(X, X)
if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_in_gggga₅(N1, B, A, C, M2))
if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_in_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L), L1, ._2₂(H, R)) -> if_append_3_in_1_gga₅(H, L, L1, R, append_3_in_gga₃(L, L1, R))
if_append_3_in_1_gga₅(H, L, L1, R, append_3_out_gga₃(L, L1, R)) -> append_3_out_gga₃(._2₂(H, L), L1, ._2₂(H, R))
if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_in_gga₃(T, M2, M))
if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_out_gga₃(T, M2, M)) -> shanoi_5_out_gggga₅(s_1₁(s_1₁(X)), A, B, C, M)

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:

shanoi_5_in_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0)) -> shanoi_5_out_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0))
shanoi_5_in_gggga₅(s_1₁(s_1₁(X)), A, B, C, M) -> if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
eq_2_in_ag₂(X, X) -> eq_2_out_ag₂(X, X)
if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_in_gggga₅(N1, B, A, C, M2))
if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_in_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L), L1, ._2₂(H, R)) -> if_append_3_in_1_gga₅(H, L, L1, R, append_3_in_gga₃(L, L1, R))
if_append_3_in_1_gga₅(H, L, L1, R, append_3_out_gga₃(L, L1, R)) -> append_3_out_gga₃(._2₂(H, L), L1, ._2₂(H, R))
if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_in_gga₃(T, M2, M))
if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_out_gga₃(T, M2, M)) -> shanoi_5_out_gggga₅(s_1₁(s_1₁(X)), A, B, C, M)

SHANOI_5_IN_GGGGA₅(s_1₁(s_1₁(X)), A, B, C, M) -> IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
SHANOI_5_IN_GGGGA₅(s_1₁(s_1₁(X)), A, B, C, M) -> EQ_2_IN_AG₂(N1, s_1₁(X))
IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> IF_SHANOI_5_IN_2_GGGGA₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> SHANOI_5_IN_GGGGA₅(N1, A, C, B, M1)
IF_SHANOI_5_IN_2_GGGGA₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> IF_SHANOI_5_IN_3_GGGGA₈(X, A, B, C, M, N1, M1, shanoi_5_in_gggga₅(N1, B, A, C, M2))
IF_SHANOI_5_IN_2_GGGGA₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> SHANOI_5_IN_GGGGA₅(N1, B, A, C, M2)
IF_SHANOI_5_IN_3_GGGGA₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> IF_SHANOI_5_IN_4_GGGGA₈(X, A, B, C, M, M1, M2, append_3_in_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T))
IF_SHANOI_5_IN_3_GGGGA₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> APPEND_3_IN_GGA₃(M1, ._2₂(mv_2₂(A, C), []_0), T)
APPEND_3_IN_GGA₃(._2₂(H, L), L1, ._2₂(H, R)) -> IF_APPEND_3_IN_1_GGA₅(H, L, L1, R, append_3_in_gga₃(L, L1, R))
APPEND_3_IN_GGA₃(._2₂(H, L), L1, ._2₂(H, R)) -> APPEND_3_IN_GGA₃(L, L1, R)
IF_SHANOI_5_IN_4_GGGGA₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> IF_SHANOI_5_IN_5_GGGGA₈(X, A, B, C, M, M2, T, append_3_in_gga₃(T, M2, M))
IF_SHANOI_5_IN_4_GGGGA₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> APPEND_3_IN_GGA₃(T, M2, M)

The TRS R consists of the following rules:

shanoi_5_in_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0)) -> shanoi_5_out_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0))
shanoi_5_in_gggga₅(s_1₁(s_1₁(X)), A, B, C, M) -> if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
eq_2_in_ag₂(X, X) -> eq_2_out_ag₂(X, X)
if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_in_gggga₅(N1, B, A, C, M2))
if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_in_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L), L1, ._2₂(H, R)) -> if_append_3_in_1_gga₅(H, L, L1, R, append_3_in_gga₃(L, L1, R))
if_append_3_in_1_gga₅(H, L, L1, R, append_3_out_gga₃(L, L1, R)) -> append_3_out_gga₃(._2₂(H, L), L1, ._2₂(H, R))
if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_in_gga₃(T, M2, M))
if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_out_gga₃(T, M2, M)) -> shanoi_5_out_gggga₅(s_1₁(s_1₁(X)), A, B, C, M)

The argument filtering Pi contains the following mapping:
shanoi_5_in_gggga₅(x1, x2, x3, x4, x5) = shanoi_5_in_gggga₄(x1, x2, x3, x4)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
._2₂(x1, x2) = ._2₂(x1, x2)
mv_2₂(x1, x2) = mv_2₂(x1, x2)
[]_0 = []_0
shanoi_5_out_gggga₅(x1, x2, x3, x4, x5) = shanoi_5_out_gggga₁(x5)
if_shanoi_5_in_1_gggga₆(x1, x2, x3, x4, x5, x6) = if_shanoi_5_in_1_gggga₄(x2, x3, x4, x6)
eq_2_in_ag₂(x1, x2) = eq_2_in_ag₁(x2)
eq_2_out_ag₂(x1, x2) = eq_2_out_ag₁(x1)
if_shanoi_5_in_2_gggga₇(x1, x2, x3, x4, x5, x6, x7) = if_shanoi_5_in_2_gggga₅(x2, x3, x4, x6, x7)
if_shanoi_5_in_3_gggga₈(x1, x2, x3, x4, x5, x6, x7, x8) = if_shanoi_5_in_3_gggga₄(x2, x4, x7, x8)
if_shanoi_5_in_4_gggga₈(x1, x2, x3, x4, x5, x6, x7, x8) = if_shanoi_5_in_4_gggga₂(x7, x8)
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_gga₂(x1, x5)
if_shanoi_5_in_5_gggga₈(x1, x2, x3, x4, x5, x6, x7, x8) = if_shanoi_5_in_5_gggga₁(x8)
IF_SHANOI_5_IN_3_GGGGA₈(x1, x2, x3, x4, x5, x6, x7, x8) = IF_SHANOI_5_IN_3_GGGGA₄(x2, x4, x7, x8)
IF_APPEND_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_APPEND_3_IN_1_GGA₂(x1, x5)
EQ_2_IN_AG₂(x1, x2) = EQ_2_IN_AG₁(x2)
SHANOI_5_IN_GGGGA₅(x1, x2, x3, x4, x5) = SHANOI_5_IN_GGGGA₄(x1, x2, x3, x4)
IF_SHANOI_5_IN_5_GGGGA₈(x1, x2, x3, x4, x5, x6, x7, x8) = IF_SHANOI_5_IN_5_GGGGA₁(x8)
IF_SHANOI_5_IN_1_GGGGA₆(x1, x2, x3, x4, x5, x6) = IF_SHANOI_5_IN_1_GGGGA₄(x2, x3, x4, x6)
IF_SHANOI_5_IN_4_GGGGA₈(x1, x2, x3, x4, x5, x6, x7, x8) = IF_SHANOI_5_IN_4_GGGGA₂(x7, x8)
IF_SHANOI_5_IN_2_GGGGA₇(x1, x2, x3, x4, x5, x6, x7) = IF_SHANOI_5_IN_2_GGGGA₅(x2, x3, x4, x6, x7)
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:

SHANOI_5_IN_GGGGA₅(s_1₁(s_1₁(X)), A, B, C, M) -> IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
SHANOI_5_IN_GGGGA₅(s_1₁(s_1₁(X)), A, B, C, M) -> EQ_2_IN_AG₂(N1, s_1₁(X))
IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> IF_SHANOI_5_IN_2_GGGGA₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> SHANOI_5_IN_GGGGA₅(N1, A, C, B, M1)
IF_SHANOI_5_IN_2_GGGGA₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> IF_SHANOI_5_IN_3_GGGGA₈(X, A, B, C, M, N1, M1, shanoi_5_in_gggga₅(N1, B, A, C, M2))
IF_SHANOI_5_IN_2_GGGGA₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> SHANOI_5_IN_GGGGA₅(N1, B, A, C, M2)
IF_SHANOI_5_IN_3_GGGGA₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> IF_SHANOI_5_IN_4_GGGGA₈(X, A, B, C, M, M1, M2, append_3_in_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T))
IF_SHANOI_5_IN_3_GGGGA₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> APPEND_3_IN_GGA₃(M1, ._2₂(mv_2₂(A, C), []_0), T)
APPEND_3_IN_GGA₃(._2₂(H, L), L1, ._2₂(H, R)) -> IF_APPEND_3_IN_1_GGA₅(H, L, L1, R, append_3_in_gga₃(L, L1, R))
APPEND_3_IN_GGA₃(._2₂(H, L), L1, ._2₂(H, R)) -> APPEND_3_IN_GGA₃(L, L1, R)
IF_SHANOI_5_IN_4_GGGGA₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> IF_SHANOI_5_IN_5_GGGGA₈(X, A, B, C, M, M2, T, append_3_in_gga₃(T, M2, M))
IF_SHANOI_5_IN_4_GGGGA₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> APPEND_3_IN_GGA₃(T, M2, M)

The TRS R consists of the following rules:

shanoi_5_in_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0)) -> shanoi_5_out_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0))
shanoi_5_in_gggga₅(s_1₁(s_1₁(X)), A, B, C, M) -> if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
eq_2_in_ag₂(X, X) -> eq_2_out_ag₂(X, X)
if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_in_gggga₅(N1, B, A, C, M2))
if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_in_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L), L1, ._2₂(H, R)) -> if_append_3_in_1_gga₅(H, L, L1, R, append_3_in_gga₃(L, L1, R))
if_append_3_in_1_gga₅(H, L, L1, R, append_3_out_gga₃(L, L1, R)) -> append_3_out_gga₃(._2₂(H, L), L1, ._2₂(H, R))
if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_in_gga₃(T, M2, M))
if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_out_gga₃(T, M2, M)) -> shanoi_5_out_gggga₅(s_1₁(s_1₁(X)), A, B, C, M)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APPEND_3_IN_GGA₃(._2₂(H, L), L1, ._2₂(H, R)) -> APPEND_3_IN_GGA₃(L, L1, R)

The TRS R consists of the following rules:

shanoi_5_in_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0)) -> shanoi_5_out_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0))
shanoi_5_in_gggga₅(s_1₁(s_1₁(X)), A, B, C, M) -> if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
eq_2_in_ag₂(X, X) -> eq_2_out_ag₂(X, X)
if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_in_gggga₅(N1, B, A, C, M2))
if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_in_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L), L1, ._2₂(H, R)) -> if_append_3_in_1_gga₅(H, L, L1, R, append_3_in_gga₃(L, L1, R))
if_append_3_in_1_gga₅(H, L, L1, R, append_3_out_gga₃(L, L1, R)) -> append_3_out_gga₃(._2₂(H, L), L1, ._2₂(H, R))
if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_in_gga₃(T, M2, M))
if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_out_gga₃(T, M2, M)) -> shanoi_5_out_gggga₅(s_1₁(s_1₁(X)), A, B, C, M)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

APPEND_3_IN_GGA₃(._2₂(H, L), L1, ._2₂(H, R)) -> APPEND_3_IN_GGA₃(L, L1, R)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

APPEND_3_IN_GGA₂(._2₂(H, L), L1) -> APPEND_3_IN_GGA₂(L, L1)

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(._2₂(H, L), L1) -> APPEND_3_IN_GGA₂(L, L1)
The graph contains the following edges 1 > 1, 2 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

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

SHANOI_5_IN_GGGGA₅(s_1₁(s_1₁(X)), A, B, C, M) -> IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
IF_SHANOI_5_IN_2_GGGGA₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> SHANOI_5_IN_GGGGA₅(N1, B, A, C, M2)
IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> IF_SHANOI_5_IN_2_GGGGA₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
IF_SHANOI_5_IN_1_GGGGA₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> SHANOI_5_IN_GGGGA₅(N1, A, C, B, M1)

The TRS R consists of the following rules:

shanoi_5_in_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0)) -> shanoi_5_out_gggga₅(s_1₁(0_0), A, B, C, ._2₂(mv_2₂(A, C), []_0))
shanoi_5_in_gggga₅(s_1₁(s_1₁(X)), A, B, C, M) -> if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_in_ag₂(N1, s_1₁(X)))
eq_2_in_ag₂(X, X) -> eq_2_out_ag₂(X, X)
if_shanoi_5_in_1_gggga₆(X, A, B, C, M, eq_2_out_ag₂(N1, s_1₁(X))) -> if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_in_gggga₅(N1, A, C, B, M1))
if_shanoi_5_in_2_gggga₇(X, A, B, C, M, N1, shanoi_5_out_gggga₅(N1, A, C, B, M1)) -> if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_in_gggga₅(N1, B, A, C, M2))
if_shanoi_5_in_3_gggga₈(X, A, B, C, M, N1, M1, shanoi_5_out_gggga₅(N1, B, A, C, M2)) -> if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_in_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T))
append_3_in_gga₃([]_0, L, L) -> append_3_out_gga₃([]_0, L, L)
append_3_in_gga₃(._2₂(H, L), L1, ._2₂(H, R)) -> if_append_3_in_1_gga₅(H, L, L1, R, append_3_in_gga₃(L, L1, R))
if_append_3_in_1_gga₅(H, L, L1, R, append_3_out_gga₃(L, L1, R)) -> append_3_out_gga₃(._2₂(H, L), L1, ._2₂(H, R))
if_shanoi_5_in_4_gggga₈(X, A, B, C, M, M1, M2, append_3_out_gga₃(M1, ._2₂(mv_2₂(A, C), []_0), T)) -> if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_in_gga₃(T, M2, M))
if_shanoi_5_in_5_gggga₈(X, A, B, C, M, M2, T, append_3_out_gga₃(T, M2, M)) -> shanoi_5_out_gggga₅(s_1₁(s_1₁(X)), A, B, C, M)

The argument filtering Pi contains the following mapping:
shanoi_5_in_gggga₅(x1, x2, x3, x4, x5) = shanoi_5_in_gggga₄(x1, x2, x3, x4)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
._2₂(x1, x2) = ._2₂(x1, x2)
mv_2₂(x1, x2) = mv_2₂(x1, x2)
[]_0 = []_0
shanoi_5_out_gggga₅(x1, x2, x3, x4, x5) = shanoi_5_out_gggga₁(x5)
if_shanoi_5_in_1_gggga₆(x1, x2, x3, x4, x5, x6) = if_shanoi_5_in_1_gggga₄(x2, x3, x4, x6)
eq_2_in_ag₂(x1, x2) = eq_2_in_ag₁(x2)
eq_2_out_ag₂(x1, x2) = eq_2_out_ag₁(x1)
if_shanoi_5_in_2_gggga₇(x1, x2, x3, x4, x5, x6, x7) = if_shanoi_5_in_2_gggga₅(x2, x3, x4, x6, x7)
if_shanoi_5_in_3_gggga₈(x1, x2, x3, x4, x5, x6, x7, x8) = if_shanoi_5_in_3_gggga₄(x2, x4, x7, x8)
if_shanoi_5_in_4_gggga₈(x1, x2, x3, x4, x5, x6, x7, x8) = if_shanoi_5_in_4_gggga₂(x7, x8)
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_gga₂(x1, x5)
if_shanoi_5_in_5_gggga₈(x1, x2, x3, x4, x5, x6, x7, x8) = if_shanoi_5_in_5_gggga₁(x8)
SHANOI_5_IN_GGGGA₅(x1, x2, x3, x4, x5) = SHANOI_5_IN_GGGGA₄(x1, x2, x3, x4)
IF_SHANOI_5_IN_1_GGGGA₆(x1, x2, x3, x4, x5, x6) = IF_SHANOI_5_IN_1_GGGGA₄(x2, x3, x4, x6)
IF_SHANOI_5_IN_2_GGGGA₇(x1, x2, x3, x4, x5, x6, x7) = IF_SHANOI_5_IN_2_GGGGA₅(x2, x3, x4, x6, x7)

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

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPPoloProof

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

SHANOI_5_IN_GGGGA₄(s_1₁(s_1₁(X)), A, B, C) -> IF_SHANOI_5_IN_1_GGGGA₄(A, B, C, eq_2_in_ag₁(s_1₁(X)))
IF_SHANOI_5_IN_2_GGGGA₅(A, B, C, N1, shanoi_5_out_gggga₁(M1)) -> SHANOI_5_IN_GGGGA₄(N1, B, A, C)
IF_SHANOI_5_IN_1_GGGGA₄(A, B, C, eq_2_out_ag₁(N1)) -> IF_SHANOI_5_IN_2_GGGGA₅(A, B, C, N1, shanoi_5_in_gggga₄(N1, A, C, B))
IF_SHANOI_5_IN_1_GGGGA₄(A, B, C, eq_2_out_ag₁(N1)) -> SHANOI_5_IN_GGGGA₄(N1, A, C, B)

The TRS R consists of the following rules:

shanoi_5_in_gggga₄(s_1₁(0_0), A, B, C) -> shanoi_5_out_gggga₁(._2₂(mv_2₂(A, C), []_0))
shanoi_5_in_gggga₄(s_1₁(s_1₁(X)), A, B, C) -> if_shanoi_5_in_1_gggga₄(A, B, C, eq_2_in_ag₁(s_1₁(X)))
eq_2_in_ag₁(X) -> eq_2_out_ag₁(X)
if_shanoi_5_in_1_gggga₄(A, B, C, eq_2_out_ag₁(N1)) -> if_shanoi_5_in_2_gggga₅(A, B, C, N1, shanoi_5_in_gggga₄(N1, A, C, B))
if_shanoi_5_in_2_gggga₅(A, B, C, N1, shanoi_5_out_gggga₁(M1)) -> if_shanoi_5_in_3_gggga₄(A, C, M1, shanoi_5_in_gggga₄(N1, B, A, C))
if_shanoi_5_in_3_gggga₄(A, C, M1, shanoi_5_out_gggga₁(M2)) -> if_shanoi_5_in_4_gggga₂(M2, append_3_in_gga₂(M1, ._2₂(mv_2₂(A, C), []_0)))
append_3_in_gga₂([]_0, L) -> append_3_out_gga₁(L)
append_3_in_gga₂(._2₂(H, L), L1) -> if_append_3_in_1_gga₂(H, append_3_in_gga₂(L, L1))
if_append_3_in_1_gga₂(H, append_3_out_gga₁(R)) -> append_3_out_gga₁(._2₂(H, R))
if_shanoi_5_in_4_gggga₂(M2, append_3_out_gga₁(T)) -> if_shanoi_5_in_5_gggga₁(append_3_in_gga₂(T, M2))
if_shanoi_5_in_5_gggga₁(append_3_out_gga₁(M)) -> shanoi_5_out_gggga₁(M)

The set Q consists of the following terms:

shanoi_5_in_gggga₄(x0, x1, x2, x3)
eq_2_in_ag₁(x0)
if_shanoi_5_in_1_gggga₄(x0, x1, x2, x3)
if_shanoi_5_in_2_gggga₅(x0, x1, x2, x3, x4)
if_shanoi_5_in_3_gggga₄(x0, x1, x2, x3)
append_3_in_gga₂(x0, x1)
if_append_3_in_1_gga₂(x0, x1)
if_shanoi_5_in_4_gggga₂(x0, x1)
if_shanoi_5_in_5_gggga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_SHANOI_5_IN_1_GGGGA₄, SHANOI_5_IN_GGGGA₄, IF_SHANOI_5_IN_2_GGGGA₅}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

SHANOI_5_IN_GGGGA₄(s_1₁(s_1₁(X)), A, B, C) -> IF_SHANOI_5_IN_1_GGGGA₄(A, B, C, eq_2_in_ag₁(s_1₁(X)))

The remaining Dependency Pairs were at least non-strictly be oriented.

IF_SHANOI_5_IN_2_GGGGA₅(A, B, C, N1, shanoi_5_out_gggga₁(M1)) -> SHANOI_5_IN_GGGGA₄(N1, B, A, C)
IF_SHANOI_5_IN_1_GGGGA₄(A, B, C, eq_2_out_ag₁(N1)) -> IF_SHANOI_5_IN_2_GGGGA₅(A, B, C, N1, shanoi_5_in_gggga₄(N1, A, C, B))
IF_SHANOI_5_IN_1_GGGGA₄(A, B, C, eq_2_out_ag₁(N1)) -> SHANOI_5_IN_GGGGA₄(N1, A, C, B)

With the implicit AFS we had to orient the following set of usable rules non-strictly.

eq_2_in_ag₁(X) -> eq_2_out_ag₁(X)

Used ordering: POLO with Polynomial interpretation:

POL(append_3_out_gga₁(x₁)) = 0
POL(0_0) = 0
POL(eq_2_in_ag₁(x₁)) = x₁
POL(if_append_3_in_1_gga₂(x₁, x₂)) = 0
POL(SHANOI_5_IN_GGGGA₄(x₁, x₂, x₃, x₄)) = x₁
POL(shanoi_5_out_gggga₁(x₁)) = 0
POL(eq_2_out_ag₁(x₁)) = x₁
POL(if_shanoi_5_in_4_gggga₂(x₁, x₂)) = 0
POL([]_0) = 0
POL(if_shanoi_5_in_1_gggga₄(x₁, x₂, x₃, x₄)) = 0
POL(if_shanoi_5_in_2_gggga₅(x₁, x₂, x₃, x₄, x₅)) = 0
POL(if_shanoi_5_in_5_gggga₁(x₁)) = 0
POL(._2₂(x₁, x₂)) = 0
POL(IF_SHANOI_5_IN_2_GGGGA₅(x₁, x₂, x₃, x₄, x₅)) = x₄
POL(if_shanoi_5_in_3_gggga₄(x₁, x₂, x₃, x₄)) = 0
POL(IF_SHANOI_5_IN_1_GGGGA₄(x₁, x₂, x₃, x₄)) = x₄
POL(shanoi_5_in_gggga₄(x₁, x₂, x₃, x₄)) = 0
POL(s_1₁(x₁)) = 1 + x₁
POL(append_3_in_gga₂(x₁, x₂)) = 0
POL(mv_2₂(x₁, x₂)) = 0

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPPoloProof

                      ↳ QDP

                        ↳ DependencyGraphProof

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

IF_SHANOI_5_IN_2_GGGGA₅(A, B, C, N1, shanoi_5_out_gggga₁(M1)) -> SHANOI_5_IN_GGGGA₄(N1, B, A, C)
IF_SHANOI_5_IN_1_GGGGA₄(A, B, C, eq_2_out_ag₁(N1)) -> IF_SHANOI_5_IN_2_GGGGA₅(A, B, C, N1, shanoi_5_in_gggga₄(N1, A, C, B))
IF_SHANOI_5_IN_1_GGGGA₄(A, B, C, eq_2_out_ag₁(N1)) -> SHANOI_5_IN_GGGGA₄(N1, A, C, B)

The TRS R consists of the following rules:

shanoi_5_in_gggga₄(s_1₁(0_0), A, B, C) -> shanoi_5_out_gggga₁(._2₂(mv_2₂(A, C), []_0))
shanoi_5_in_gggga₄(s_1₁(s_1₁(X)), A, B, C) -> if_shanoi_5_in_1_gggga₄(A, B, C, eq_2_in_ag₁(s_1₁(X)))
eq_2_in_ag₁(X) -> eq_2_out_ag₁(X)
if_shanoi_5_in_1_gggga₄(A, B, C, eq_2_out_ag₁(N1)) -> if_shanoi_5_in_2_gggga₅(A, B, C, N1, shanoi_5_in_gggga₄(N1, A, C, B))
if_shanoi_5_in_2_gggga₅(A, B, C, N1, shanoi_5_out_gggga₁(M1)) -> if_shanoi_5_in_3_gggga₄(A, C, M1, shanoi_5_in_gggga₄(N1, B, A, C))
if_shanoi_5_in_3_gggga₄(A, C, M1, shanoi_5_out_gggga₁(M2)) -> if_shanoi_5_in_4_gggga₂(M2, append_3_in_gga₂(M1, ._2₂(mv_2₂(A, C), []_0)))
append_3_in_gga₂([]_0, L) -> append_3_out_gga₁(L)
append_3_in_gga₂(._2₂(H, L), L1) -> if_append_3_in_1_gga₂(H, append_3_in_gga₂(L, L1))
if_append_3_in_1_gga₂(H, append_3_out_gga₁(R)) -> append_3_out_gga₁(._2₂(H, R))
if_shanoi_5_in_4_gggga₂(M2, append_3_out_gga₁(T)) -> if_shanoi_5_in_5_gggga₁(append_3_in_gga₂(T, M2))
if_shanoi_5_in_5_gggga₁(append_3_out_gga₁(M)) -> shanoi_5_out_gggga₁(M)

The set Q consists of the following terms:

shanoi_5_in_gggga₄(x0, x1, x2, x3)
eq_2_in_ag₁(x0)
if_shanoi_5_in_1_gggga₄(x0, x1, x2, x3)
if_shanoi_5_in_2_gggga₅(x0, x1, x2, x3, x4)
if_shanoi_5_in_3_gggga₄(x0, x1, x2, x3)
append_3_in_gga₂(x0, x1)
if_append_3_in_1_gga₂(x0, x1)
if_shanoi_5_in_4_gggga₂(x0, x1)
if_shanoi_5_in_5_gggga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {SHANOI_5_IN_GGGGA₄, IF_SHANOI_5_IN_2_GGGGA₅, IF_SHANOI_5_IN_1_GGGGA₄}.
The approximation of the Dependency Graph contains 0 SCCs with 3 less nodes.