Left Termination of the query pattern merge_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇔)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇔)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
      ↳21 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
  ↳26 PrologToPiTRSProof (⇐)
    ↳27 PiTRS
    ↳28 DependencyPairsProof (⇔)
    ↳29 PiDP
    ↳30 DependencyGraphProof (⇔)
    ↳31 AND
      ↳32 PiDP
        ↳33 UsableRulesProof (⇔)
        ↳34 PiDP
        ↳35 PiDPToQDPProof (⇔)
        ↳36 QDP
        ↳37 QDPSizeChangeProof (⇔)
        ↳38 TRUE
      ↳39 PiDP
        ↳40 UsableRulesProof (⇔)
        ↳41 PiDP
        ↳42 PiDPToQDPProof (⇔)
        ↳43 QDP
        ↳44 QDPSizeChangeProof (⇔)
        ↳45 TRUE
      ↳46 PiDP
        ↳47 UsableRulesProof (⇔)
        ↳48 PiDP
        ↳49 PiDPToQDPProof (⇐)
        ↳50 QDP
        ↳51 MRRProof (⇔)
        ↳52 QDP
        ↳53 DependencyGraphProof (⇔)
        ↳54 TRUE

(0) Obligation:

Clauses:

merge([], X, X).
merge(X, [], X).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs)).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs)).
less(0, s(0)).
less(s(X), s(Y)) :- less(X, Y).
leq(0, 0).
leq(0, s(0)).
leq(s(X), s(Y)) :- leq(X, Y).

Queries:

merge(g,g,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
merge_in: (b,b,f)
leq_in: (b,b)
less_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x3, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x3, x6)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LEQ_IN_GG(X, Y)
LEQ_IN_GG(s(X), s(Y)) → U6_GG(X, Y, leq_in_gg(X, Y))
LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GG(Y, X)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x3, x6)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
LEQ_IN_GG(x1, x2)  =  LEQ_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x3)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x3, x6)

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

(4) Obligation:

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

MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LEQ_IN_GG(X, Y)
LEQ_IN_GG(s(X), s(Y)) → U6_GG(X, Y, leq_in_gg(X, Y))
LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GG(Y, X)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x3, x6)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
LEQ_IN_GG(x1, x2)  =  LEQ_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x3)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x3, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(13) TRUE

(14) Obligation:

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x3, x6)
LEQ_IN_GG(x1, x2)  =  LEQ_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(20) TRUE

(21) Obligation:

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

U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x3, x6)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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

U1_GGA(X, Xs, Y, Ys, leq_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U1_GGA(X, Xs, Y, Ys, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U3_GGA(X, Xs, Y, Ys, less_in_gg(Y, X))
U3_GGA(X, Xs, Y, Ys, less_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)

The TRS R consists of the following rules:

leq_in_gg(0, 0) → leq_out_gg
leq_in_gg(0, s(0)) → leq_out_gg
leq_in_gg(s(X), s(Y)) → U6_gg(leq_in_gg(X, Y))
less_in_gg(0, s(0)) → less_out_gg
less_in_gg(s(X), s(Y)) → U5_gg(less_in_gg(X, Y))
U6_gg(leq_out_gg) → leq_out_gg
U5_gg(less_out_gg) → less_out_gg

The set Q consists of the following terms:

leq_in_gg(x0, x1)
less_in_gg(x0, x1)
U6_gg(x0)
U5_gg(x0)

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

(26) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
merge_in: (b,b,f)
leq_in: (b,b)
less_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg(x1, x2)
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(27) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg(x1, x2)
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)

(28) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LEQ_IN_GG(X, Y)
LEQ_IN_GG(s(X), s(Y)) → U6_GG(X, Y, leq_in_gg(X, Y))
LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GG(Y, X)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg(x1, x2)
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
LEQ_IN_GG(x1, x2)  =  LEQ_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)

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

(29) Obligation:

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

MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LEQ_IN_GG(X, Y)
LEQ_IN_GG(s(X), s(Y)) → U6_GG(X, Y, leq_in_gg(X, Y))
LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GG(Y, X)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg(x1, x2)
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
LEQ_IN_GG(x1, x2)  =  LEQ_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)

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

(30) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes.

(31) Complex Obligation (AND)

(32) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg(x1, x2)
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(35) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(36) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(37) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(38) TRUE

(39) Obligation:

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg(x1, x2)
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
LEQ_IN_GG(x1, x2)  =  LEQ_IN_GG(x1, x2)

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

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) Obligation:

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

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

(42) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(43) Obligation:

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(44) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(45) TRUE

(46) Obligation:

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

U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
[]  =  []
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg(x1, x2)
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)

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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) Obligation:

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

U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
leq_in_gg(x1, x2)  =  leq_in_gg(x1, x2)
0  =  0
leq_out_gg(x1, x2)  =  leq_out_gg(x1, x2)
s(x1)  =  s(x1)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)

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

(49) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(50) Obligation:

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

U1_GGA(X, Xs, Y, Ys, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U1_GGA(X, Xs, Y, Ys, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U3_GGA(X, Xs, Y, Ys, less_in_gg(Y, X))
U3_GGA(X, Xs, Y, Ys, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys)

The TRS R consists of the following rules:

leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The set Q consists of the following terms:

leq_in_gg(x0, x1)
less_in_gg(x0, x1)
U6_gg(x0, x1, x2)
U5_gg(x0, x1, x2)

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

(51) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U1_GGA(X, Xs, Y, Ys, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U3_GGA(X, Xs, Y, Ys, less_in_gg(Y, X))


Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + 2·x1 + x2   
POL(0) = 0   
POL(MERGE_IN_GGA(x1, x2)) = 2·x1 + 2·x2   
POL(U1_GGA(x1, x2, x3, x4, x5)) = 2 + x1 + 2·x2 + 2·x3 + 2·x4 + x5   
POL(U3_GGA(x1, x2, x3, x4, x5)) = 2 + 2·x1 + 2·x2 + 2·x3 + 2·x4 + 2·x5   
POL(U5_gg(x1, x2, x3)) = x1 + x2 + x3   
POL(U6_gg(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(leq_in_gg(x1, x2)) = 2·x1 + 2·x2   
POL(leq_out_gg(x1, x2)) = 2·x1 + 2·x2   
POL(less_in_gg(x1, x2)) = x1 + x2   
POL(less_out_gg(x1, x2)) = x1 + x2   
POL(s(x1)) = 2·x1   

(52) Obligation:

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

U1_GGA(X, Xs, Y, Ys, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys))
U3_GGA(X, Xs, Y, Ys, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys)

The TRS R consists of the following rules:

leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The set Q consists of the following terms:

leq_in_gg(x0, x1)
less_in_gg(x0, x1)
U6_gg(x0, x1, x2)
U5_gg(x0, x1, x2)

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

(53) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(54) TRUE