Left Termination proof of /tmp/tmpCdEKsa/merge.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

merge₃({}₀, X, X).
merge₃(X, {}₀, X).
merge₃(.₂(X, Xs), .₂(Y, Ys), .₂(X, Zs)) :- leq₂(X, Y), merge₃(Xs, .₂(Y, Ys), Zs).
merge₃(.₂(X, Xs), .₂(Y, Ys), .₂(Y, Zs)) :- less₂(Y, X), merge₃(.₂(X, Xs), Ys, Zs).
less₂(0₀, s₁(0₀)).
less₂(s₁(X), s₁(Y)) :- less₂(X, Y).
leq₂(0₀, 0₀).
leq₂(0₀, s₁(0₀)).
leq₂(s₁(X), s₁(Y)) :- leq₂(X, Y).

With regard to the inferred argument filtering the predicates were used in the following modes:
merge₃: (b,b,f)
leq₂: (b,b)
less₂: (b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

merge_3_in_gga₃([]_0, X, X) -> merge_3_out_gga₃([]_0, X, X)
merge_3_in_gga₃(X, []_0, X) -> merge_3_out_gga₃(X, []_0, X)
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg₂(0_0, 0_0)
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg₂(0_0, s_1₁(0_0))
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₃(X, Y, leq_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₃(X, Y, leq_2_out_gg₂(X, Y)) -> leq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg₂(0_0, s_1₁(0_0))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(._2₂(X, Xs), Ys, Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs))
if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(Xs, ._2₂(Y, Ys), Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs))

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:

merge_3_in_gga₃([]_0, X, X) -> merge_3_out_gga₃([]_0, X, X)
merge_3_in_gga₃(X, []_0, X) -> merge_3_out_gga₃(X, []_0, X)
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg₂(0_0, 0_0)
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg₂(0_0, s_1₁(0_0))
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₃(X, Y, leq_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₃(X, Y, leq_2_out_gg₂(X, Y)) -> leq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg₂(0_0, s_1₁(0_0))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(._2₂(X, Xs), Ys, Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs))
if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(Xs, ._2₂(Y, Ys), Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs))

MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> LEQ_2_IN_GG₂(X, Y)
LEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LEQ_2_IN_1_GG₃(X, Y, leq_2_in_gg₂(X, Y))
LEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LEQ_2_IN_GG₂(X, Y)
IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> IF_MERGE_3_IN_2_GGA₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> MERGE_3_IN_GGA₃(Xs, ._2₂(Y, Ys), Zs)
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> LESS_2_IN_GG₂(Y, X)
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_GG₃(X, Y, less_2_in_gg₂(X, Y))
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)
IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> IF_MERGE_3_IN_4_GGA₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> MERGE_3_IN_GGA₃(._2₂(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_3_in_gga₃([]_0, X, X) -> merge_3_out_gga₃([]_0, X, X)
merge_3_in_gga₃(X, []_0, X) -> merge_3_out_gga₃(X, []_0, X)
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg₂(0_0, 0_0)
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg₂(0_0, s_1₁(0_0))
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₃(X, Y, leq_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₃(X, Y, leq_2_out_gg₂(X, Y)) -> leq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg₂(0_0, s_1₁(0_0))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(._2₂(X, Xs), Ys, Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs))
if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(Xs, ._2₂(Y, Ys), Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs))

The argument filtering Pi contains the following mapping:
merge_3_in_gga₃(x1, x2, x3) = merge_3_in_gga₂(x1, x2)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
merge_3_out_gga₃(x1, x2, x3) = merge_3_out_gga₁(x3)
if_merge_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_merge_3_in_1_gga₅(x1, x2, x3, x4, x6)
leq_2_in_gg₂(x1, x2) = leq_2_in_gg₂(x1, x2)
leq_2_out_gg₂(x1, x2) = leq_2_out_gg
if_leq_2_in_1_gg₃(x1, x2, x3) = if_leq_2_in_1_gg₁(x3)
if_merge_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_merge_3_in_2_gga₂(x1, x6)
if_merge_3_in_3_gga₆(x1, x2, x3, x4, x5, x6) = if_merge_3_in_3_gga₅(x1, x2, x3, x4, x6)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
if_merge_3_in_4_gga₆(x1, x2, x3, x4, x5, x6) = if_merge_3_in_4_gga₂(x3, x6)
IF_MERGE_3_IN_2_GGA₆(x1, x2, x3, x4, x5, x6) = IF_MERGE_3_IN_2_GGA₂(x1, x6)
MERGE_3_IN_GGA₃(x1, x2, x3) = MERGE_3_IN_GGA₂(x1, x2)
IF_MERGE_3_IN_1_GGA₆(x1, x2, x3, x4, x5, x6) = IF_MERGE_3_IN_1_GGA₅(x1, x2, x3, x4, x6)
IF_MERGE_3_IN_3_GGA₆(x1, x2, x3, x4, x5, x6) = IF_MERGE_3_IN_3_GGA₅(x1, x2, x3, x4, x6)
IF_MERGE_3_IN_4_GGA₆(x1, x2, x3, x4, x5, x6) = IF_MERGE_3_IN_4_GGA₂(x3, x6)
IF_LESS_2_IN_1_GG₃(x1, x2, x3) = IF_LESS_2_IN_1_GG₁(x3)
LEQ_2_IN_GG₂(x1, x2) = LEQ_2_IN_GG₂(x1, x2)
IF_LEQ_2_IN_1_GG₃(x1, x2, x3) = IF_LEQ_2_IN_1_GG₁(x3)
LESS_2_IN_GG₂(x1, x2) = LESS_2_IN_GG₂(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:

MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> LEQ_2_IN_GG₂(X, Y)
LEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LEQ_2_IN_1_GG₃(X, Y, leq_2_in_gg₂(X, Y))
LEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LEQ_2_IN_GG₂(X, Y)
IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> IF_MERGE_3_IN_2_GGA₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> MERGE_3_IN_GGA₃(Xs, ._2₂(Y, Ys), Zs)
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> LESS_2_IN_GG₂(Y, X)
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_GG₃(X, Y, less_2_in_gg₂(X, Y))
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)
IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> IF_MERGE_3_IN_4_GGA₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> MERGE_3_IN_GGA₃(._2₂(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_3_in_gga₃([]_0, X, X) -> merge_3_out_gga₃([]_0, X, X)
merge_3_in_gga₃(X, []_0, X) -> merge_3_out_gga₃(X, []_0, X)
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg₂(0_0, 0_0)
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg₂(0_0, s_1₁(0_0))
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₃(X, Y, leq_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₃(X, Y, leq_2_out_gg₂(X, Y)) -> leq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg₂(0_0, s_1₁(0_0))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(._2₂(X, Xs), Ys, Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs))
if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(Xs, ._2₂(Y, Ys), Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)

The TRS R consists of the following rules:

merge_3_in_gga₃([]_0, X, X) -> merge_3_out_gga₃([]_0, X, X)
merge_3_in_gga₃(X, []_0, X) -> merge_3_out_gga₃(X, []_0, X)
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg₂(0_0, 0_0)
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg₂(0_0, s_1₁(0_0))
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₃(X, Y, leq_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₃(X, Y, leq_2_out_gg₂(X, Y)) -> leq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg₂(0_0, s_1₁(0_0))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(._2₂(X, Xs), Ys, Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs))
if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(Xs, ._2₂(Y, Ys), Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)

R is empty.
Pi is empty.
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:

LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LESS_2_IN_GG₂}.
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:

LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

LEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LEQ_2_IN_GG₂(X, Y)

The TRS R consists of the following rules:

merge_3_in_gga₃([]_0, X, X) -> merge_3_out_gga₃([]_0, X, X)
merge_3_in_gga₃(X, []_0, X) -> merge_3_out_gga₃(X, []_0, X)
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg₂(0_0, 0_0)
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg₂(0_0, s_1₁(0_0))
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₃(X, Y, leq_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₃(X, Y, leq_2_out_gg₂(X, Y)) -> leq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg₂(0_0, s_1₁(0_0))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(._2₂(X, Xs), Ys, Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs))
if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(Xs, ._2₂(Y, Ys), Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

LEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LEQ_2_IN_GG₂(X, Y)

R is empty.
Pi is empty.
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:

LEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LEQ_2_IN_GG₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LEQ_2_IN_GG₂}.
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:

LEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LEQ_2_IN_GG₂(X, Y)
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:

IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> MERGE_3_IN_GGA₃(Xs, ._2₂(Y, Ys), Zs)
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> MERGE_3_IN_GGA₃(._2₂(X, Xs), Ys, Zs)
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))

The TRS R consists of the following rules:

merge_3_in_gga₃([]_0, X, X) -> merge_3_out_gga₃([]_0, X, X)
merge_3_in_gga₃(X, []_0, X) -> merge_3_out_gga₃(X, []_0, X)
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg₂(0_0, 0_0)
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg₂(0_0, s_1₁(0_0))
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₃(X, Y, leq_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₃(X, Y, leq_2_out_gg₂(X, Y)) -> leq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_1_gga₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(Xs, ._2₂(Y, Ys), Zs))
merge_3_in_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg₂(0_0, s_1₁(0_0))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_merge_3_in_3_gga₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_in_gga₃(._2₂(X, Xs), Ys, Zs))
if_merge_3_in_4_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(._2₂(X, Xs), Ys, Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs))
if_merge_3_in_2_gga₆(X, Xs, Y, Ys, Zs, merge_3_out_gga₃(Xs, ._2₂(Y, Ys), Zs)) -> merge_3_out_gga₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs))

The argument filtering Pi contains the following mapping:
merge_3_in_gga₃(x1, x2, x3) = merge_3_in_gga₂(x1, x2)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
merge_3_out_gga₃(x1, x2, x3) = merge_3_out_gga₁(x3)
if_merge_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_merge_3_in_1_gga₅(x1, x2, x3, x4, x6)
leq_2_in_gg₂(x1, x2) = leq_2_in_gg₂(x1, x2)
leq_2_out_gg₂(x1, x2) = leq_2_out_gg
if_leq_2_in_1_gg₃(x1, x2, x3) = if_leq_2_in_1_gg₁(x3)
if_merge_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_merge_3_in_2_gga₂(x1, x6)
if_merge_3_in_3_gga₆(x1, x2, x3, x4, x5, x6) = if_merge_3_in_3_gga₅(x1, x2, x3, x4, x6)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
if_merge_3_in_4_gga₆(x1, x2, x3, x4, x5, x6) = if_merge_3_in_4_gga₂(x3, x6)
MERGE_3_IN_GGA₃(x1, x2, x3) = MERGE_3_IN_GGA₂(x1, x2)
IF_MERGE_3_IN_1_GGA₆(x1, x2, x3, x4, x5, x6) = IF_MERGE_3_IN_1_GGA₅(x1, x2, x3, x4, x6)
IF_MERGE_3_IN_3_GGA₆(x1, x2, x3, x4, x5, x6) = IF_MERGE_3_IN_3_GGA₅(x1, x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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:

IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_out_gg₂(X, Y)) -> MERGE_3_IN_GGA₃(Xs, ._2₂(Y, Ys), Zs)
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(X, Zs)) -> IF_MERGE_3_IN_1_GGA₆(X, Xs, Y, Ys, Zs, leq_2_in_gg₂(X, Y))
IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_out_gg₂(Y, X)) -> MERGE_3_IN_GGA₃(._2₂(X, Xs), Ys, Zs)
MERGE_3_IN_GGA₃(._2₂(X, Xs), ._2₂(Y, Ys), ._2₂(Y, Zs)) -> IF_MERGE_3_IN_3_GGA₆(X, Xs, Y, Ys, Zs, less_2_in_gg₂(Y, X))

The TRS R consists of the following rules:

leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg₂(0_0, 0_0)
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg₂(0_0, s_1₁(0_0))
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₃(X, Y, leq_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg₂(0_0, s_1₁(0_0))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₃(X, Y, leq_2_out_gg₂(X, Y)) -> leq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
leq_2_in_gg₂(x1, x2) = leq_2_in_gg₂(x1, x2)
leq_2_out_gg₂(x1, x2) = leq_2_out_gg
if_leq_2_in_1_gg₃(x1, x2, x3) = if_leq_2_in_1_gg₁(x3)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
MERGE_3_IN_GGA₃(x1, x2, x3) = MERGE_3_IN_GGA₂(x1, x2)
IF_MERGE_3_IN_1_GGA₆(x1, x2, x3, x4, x5, x6) = IF_MERGE_3_IN_1_GGA₅(x1, x2, x3, x4, x6)
IF_MERGE_3_IN_3_GGA₆(x1, x2, x3, x4, x5, x6) = IF_MERGE_3_IN_3_GGA₅(x1, x2, x3, x4, x6)

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

                        ↳ QDPPoloProof

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

IF_MERGE_3_IN_1_GGA₅(X, Xs, Y, Ys, leq_2_out_gg) -> MERGE_3_IN_GGA₂(Xs, ._2₂(Y, Ys))
MERGE_3_IN_GGA₂(._2₂(X, Xs), ._2₂(Y, Ys)) -> IF_MERGE_3_IN_1_GGA₅(X, Xs, Y, Ys, leq_2_in_gg₂(X, Y))
IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_out_gg) -> MERGE_3_IN_GGA₂(._2₂(X, Xs), Ys)
MERGE_3_IN_GGA₂(._2₂(X, Xs), ._2₂(Y, Ys)) -> IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_in_gg₂(Y, X))

The TRS R consists of the following rules:

leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₁(leq_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₁(less_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₁(leq_2_out_gg) -> leq_2_out_gg
if_less_2_in_1_gg₁(less_2_out_gg) -> less_2_out_gg

The set Q consists of the following terms:

leq_2_in_gg₂(x0, x1)
less_2_in_gg₂(x0, x1)
if_leq_2_in_1_gg₁(x0)
if_less_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MERGE_3_IN_GGA₂, IF_MERGE_3_IN_1_GGA₅, IF_MERGE_3_IN_3_GGA₅}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

MERGE_3_IN_GGA₂(._2₂(X, Xs), ._2₂(Y, Ys)) -> IF_MERGE_3_IN_1_GGA₅(X, Xs, Y, Ys, leq_2_in_gg₂(X, Y))

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

IF_MERGE_3_IN_1_GGA₅(X, Xs, Y, Ys, leq_2_out_gg) -> MERGE_3_IN_GGA₂(Xs, ._2₂(Y, Ys))
IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_out_gg) -> MERGE_3_IN_GGA₂(._2₂(X, Xs), Ys)
MERGE_3_IN_GGA₂(._2₂(X, Xs), ._2₂(Y, Ys)) -> IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_in_gg₂(Y, X))

With the implicit AFS there is no usable rule.

Used ordering: POLO with Polynomial interpretation:

POL(if_leq_2_in_1_gg₁(x₁)) = 0
POL(less_2_out_gg) = 0
POL(0_0) = 0
POL(MERGE_3_IN_GGA₂(x₁, x₂)) = x₁
POL(._2₂(x₁, x₂)) = 1 + x₂
POL(IF_MERGE_3_IN_1_GGA₅(x₁, x₂, x₃, x₄, x₅)) = x₂
POL(if_less_2_in_1_gg₁(x₁)) = 0
POL(leq_2_out_gg) = 0
POL(leq_2_in_gg₂(x₁, x₂)) = 0
POL(s_1₁(x₁)) = 0
POL(IF_MERGE_3_IN_3_GGA₅(x₁, x₂, x₃, x₄, x₅)) = 1 + x₂
POL(less_2_in_gg₂(x₁, x₂)) = 0

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

IF_MERGE_3_IN_1_GGA₅(X, Xs, Y, Ys, leq_2_out_gg) -> MERGE_3_IN_GGA₂(Xs, ._2₂(Y, Ys))
IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_out_gg) -> MERGE_3_IN_GGA₂(._2₂(X, Xs), Ys)
MERGE_3_IN_GGA₂(._2₂(X, Xs), ._2₂(Y, Ys)) -> IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_in_gg₂(Y, X))

The TRS R consists of the following rules:

leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₁(leq_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₁(less_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₁(leq_2_out_gg) -> leq_2_out_gg
if_less_2_in_1_gg₁(less_2_out_gg) -> less_2_out_gg

The set Q consists of the following terms:

leq_2_in_gg₂(x0, x1)
less_2_in_gg₂(x0, x1)
if_leq_2_in_1_gg₁(x0)
if_less_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MERGE_3_IN_GGA₂, IF_MERGE_3_IN_1_GGA₅, IF_MERGE_3_IN_3_GGA₅}.
The approximation of the Dependency Graph contains 1 SCC with 1 less node.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

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

MERGE_3_IN_GGA₂(._2₂(X, Xs), ._2₂(Y, Ys)) -> IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_in_gg₂(Y, X))
IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_out_gg) -> MERGE_3_IN_GGA₂(._2₂(X, Xs), Ys)

The TRS R consists of the following rules:

leq_2_in_gg₂(0_0, 0_0) -> leq_2_out_gg
leq_2_in_gg₂(0_0, s_1₁(0_0)) -> leq_2_out_gg
leq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_leq_2_in_1_gg₁(leq_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(0_0)) -> less_2_out_gg
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₁(less_2_in_gg₂(X, Y))
if_leq_2_in_1_gg₁(leq_2_out_gg) -> leq_2_out_gg
if_less_2_in_1_gg₁(less_2_out_gg) -> less_2_out_gg

The set Q consists of the following terms:

leq_2_in_gg₂(x0, x1)
less_2_in_gg₂(x0, x1)
if_leq_2_in_1_gg₁(x0)
if_less_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_MERGE_3_IN_3_GGA₅, MERGE_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:

IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_out_gg) -> MERGE_3_IN_GGA₂(._2₂(X, Xs), Ys)
The graph contains the following edges 4 >= 2
MERGE_3_IN_GGA₂(._2₂(X, Xs), ._2₂(Y, Ys)) -> IF_MERGE_3_IN_3_GGA₅(X, Xs, Y, Ys, less_2_in_gg₂(Y, X))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4