(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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

leq33(s(T41), s(T42)) :- leq33(T41, T42).
less51(s(T67), s(T68)) :- less51(T67, T68).
merge1(.(0, T18), .(0, T20), .(0, T22)) :- merge1(T18, .(0, T20), T22).
merge1(.(0, T18), .(s(0), T20), .(0, T22)) :- merge1(T18, .(s(0), T20), T22).
merge1(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) :- leq33(T35, T36).
merge1(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) :- ','(leqc33(T35, T36), merge1(T18, .(s(T36), T20), T22)).
merge1(.(T57, T58), .(T59, T60), .(T59, T62)) :- less51(T59, T57).
merge1(.(T57, T58), .(T59, T60), .(T59, T62)) :- ','(lessc51(T59, T57), merge1(.(T57, T58), T60, T62)).
merge1(.(s(0), T79), .(0, T81), .(0, T83)) :- merge1(.(s(0), T79), T81, T83).
merge1(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) :- less51(T90, T91).
merge1(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) :- ','(lessc51(T90, T91), merge1(.(s(T91), T79), T81, T83)).

Clauses:

mergec1([], T5, T5).
mergec1([], [], []).
mergec1(T7, [], T7).
mergec1(.(0, T18), .(0, T20), .(0, T22)) :- mergec1(T18, .(0, T20), T22).
mergec1(.(0, T18), .(s(0), T20), .(0, T22)) :- mergec1(T18, .(s(0), T20), T22).
mergec1(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) :- ','(leqc33(T35, T36), mergec1(T18, .(s(T36), T20), T22)).
mergec1(.(T57, T58), .(T59, T60), .(T59, T62)) :- ','(lessc51(T59, T57), mergec1(.(T57, T58), T60, T62)).
mergec1(.(s(0), T79), .(0, T81), .(0, T83)) :- mergec1(.(s(0), T79), T81, T83).
mergec1(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) :- ','(lessc51(T90, T91), mergec1(.(s(T91), T79), T81, T83)).
leqc33(0, 0).
leqc33(0, s(0)).
leqc33(s(T41), s(T42)) :- leqc33(T41, T42).
lessc51(0, s(0)).
lessc51(s(T67), s(T68)) :- lessc51(T67, T68).

Afs:

merge1(x1, x2, x3)  =  merge1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
merge1_in: (b,b,f)
leq33_in: (b,b)
leqc33_in: (b,b)
less51_in: (b,b)
lessc51_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → U3_GGA(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(0, T20), T22)
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → U4_GGA(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(s(0), T20), T22)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_GGA(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → LEQ33_IN_GG(T35, T36)
LEQ33_IN_GG(s(T41), s(T42)) → U1_GG(T41, T42, leq33_in_gg(T41, T42))
LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U6_GGA(T35, T18, T36, T20, T22, leqc33_in_gg(T35, T36))
U6_GGA(T35, T18, T36, T20, T22, leqc33_out_gg(T35, T36)) → U7_GGA(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
U6_GGA(T35, T18, T36, T20, T22, leqc33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20), T22)
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U8_GGA(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → LESS51_IN_GG(T59, T57)
LESS51_IN_GG(s(T67), s(T68)) → U2_GG(T67, T68, less51_in_gg(T67, T68))
LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U9_GGA(T57, T58, T59, T60, T62, lessc51_in_gg(T59, T57))
U9_GGA(T57, T58, T59, T60, T62, lessc51_out_gg(T59, T57)) → U10_GGA(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
U9_GGA(T57, T58, T59, T60, T62, lessc51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60, T62)
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → U11_GGA(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → MERGE1_IN_GGA(.(s(0), T79), T81, T83)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U12_GGA(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → LESS51_IN_GG(T90, T91)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U13_GGA(T91, T79, T90, T81, T83, lessc51_in_gg(T90, T91))
U13_GGA(T91, T79, T90, T81, T83, lessc51_out_gg(T90, T91)) → U14_GGA(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U13_GGA(T91, T79, T90, T81, T83, lessc51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81, T83)

The TRS R consists of the following rules:

leqc33_in_gg(0, 0) → leqc33_out_gg(0, 0)
leqc33_in_gg(0, s(0)) → leqc33_out_gg(0, s(0))
leqc33_in_gg(s(T41), s(T42)) → U25_gg(T41, T42, leqc33_in_gg(T41, T42))
U25_gg(T41, T42, leqc33_out_gg(T41, T42)) → leqc33_out_gg(s(T41), s(T42))
lessc51_in_gg(0, s(0)) → lessc51_out_gg(0, s(0))
lessc51_in_gg(s(T67), s(T68)) → U26_gg(T67, T68, lessc51_in_gg(T67, T68))
U26_gg(T67, T68, lessc51_out_gg(T67, T68)) → lessc51_out_gg(s(T67), s(T68))

The argument filtering Pi contains the following mapping:
merge1_in_gga(x1, x2, x3)  =  merge1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
leq33_in_gg(x1, x2)  =  leq33_in_gg(x1, x2)
leqc33_in_gg(x1, x2)  =  leqc33_in_gg(x1, x2)
leqc33_out_gg(x1, x2)  =  leqc33_out_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
less51_in_gg(x1, x2)  =  less51_in_gg(x1, x2)
lessc51_in_gg(x1, x2)  =  lessc51_in_gg(x1, x2)
lessc51_out_gg(x1, x2)  =  lessc51_out_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
MERGE1_IN_GGA(x1, x2, x3)  =  MERGE1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
LEQ33_IN_GG(x1, x2)  =  LEQ33_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x1, x2, x3, x4, x6)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x2, x3, x4, x6)
LESS51_IN_GG(x1, x2)  =  LESS51_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U9_GGA(x1, x2, x3, x4, x5, x6)  =  U9_GGA(x1, x2, x3, x4, x6)
U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x2, x3, x4, x6)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4, x5, x6)  =  U12_GGA(x1, x2, x3, x4, x6)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → U3_GGA(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(0, T20), T22)
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → U4_GGA(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(s(0), T20), T22)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_GGA(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → LEQ33_IN_GG(T35, T36)
LEQ33_IN_GG(s(T41), s(T42)) → U1_GG(T41, T42, leq33_in_gg(T41, T42))
LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U6_GGA(T35, T18, T36, T20, T22, leqc33_in_gg(T35, T36))
U6_GGA(T35, T18, T36, T20, T22, leqc33_out_gg(T35, T36)) → U7_GGA(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
U6_GGA(T35, T18, T36, T20, T22, leqc33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20), T22)
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U8_GGA(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → LESS51_IN_GG(T59, T57)
LESS51_IN_GG(s(T67), s(T68)) → U2_GG(T67, T68, less51_in_gg(T67, T68))
LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U9_GGA(T57, T58, T59, T60, T62, lessc51_in_gg(T59, T57))
U9_GGA(T57, T58, T59, T60, T62, lessc51_out_gg(T59, T57)) → U10_GGA(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
U9_GGA(T57, T58, T59, T60, T62, lessc51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60, T62)
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → U11_GGA(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → MERGE1_IN_GGA(.(s(0), T79), T81, T83)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U12_GGA(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → LESS51_IN_GG(T90, T91)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U13_GGA(T91, T79, T90, T81, T83, lessc51_in_gg(T90, T91))
U13_GGA(T91, T79, T90, T81, T83, lessc51_out_gg(T90, T91)) → U14_GGA(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U13_GGA(T91, T79, T90, T81, T83, lessc51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81, T83)

The TRS R consists of the following rules:

leqc33_in_gg(0, 0) → leqc33_out_gg(0, 0)
leqc33_in_gg(0, s(0)) → leqc33_out_gg(0, s(0))
leqc33_in_gg(s(T41), s(T42)) → U25_gg(T41, T42, leqc33_in_gg(T41, T42))
U25_gg(T41, T42, leqc33_out_gg(T41, T42)) → leqc33_out_gg(s(T41), s(T42))
lessc51_in_gg(0, s(0)) → lessc51_out_gg(0, s(0))
lessc51_in_gg(s(T67), s(T68)) → U26_gg(T67, T68, lessc51_in_gg(T67, T68))
U26_gg(T67, T68, lessc51_out_gg(T67, T68)) → lessc51_out_gg(s(T67), s(T68))

The argument filtering Pi contains the following mapping:
merge1_in_gga(x1, x2, x3)  =  merge1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
leq33_in_gg(x1, x2)  =  leq33_in_gg(x1, x2)
leqc33_in_gg(x1, x2)  =  leqc33_in_gg(x1, x2)
leqc33_out_gg(x1, x2)  =  leqc33_out_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
less51_in_gg(x1, x2)  =  less51_in_gg(x1, x2)
lessc51_in_gg(x1, x2)  =  lessc51_in_gg(x1, x2)
lessc51_out_gg(x1, x2)  =  lessc51_out_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
MERGE1_IN_GGA(x1, x2, x3)  =  MERGE1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
LEQ33_IN_GG(x1, x2)  =  LEQ33_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x1, x2, x3, x4, x6)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x2, x3, x4, x6)
LESS51_IN_GG(x1, x2)  =  LESS51_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U9_GGA(x1, x2, x3, x4, x5, x6)  =  U9_GGA(x1, x2, x3, x4, x6)
U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x2, x3, x4, x6)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4, x5, x6)  =  U12_GGA(x1, x2, x3, x4, x6)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, 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 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)

The TRS R consists of the following rules:

leqc33_in_gg(0, 0) → leqc33_out_gg(0, 0)
leqc33_in_gg(0, s(0)) → leqc33_out_gg(0, s(0))
leqc33_in_gg(s(T41), s(T42)) → U25_gg(T41, T42, leqc33_in_gg(T41, T42))
U25_gg(T41, T42, leqc33_out_gg(T41, T42)) → leqc33_out_gg(s(T41), s(T42))
lessc51_in_gg(0, s(0)) → lessc51_out_gg(0, s(0))
lessc51_in_gg(s(T67), s(T68)) → U26_gg(T67, T68, lessc51_in_gg(T67, T68))
U26_gg(T67, T68, lessc51_out_gg(T67, T68)) → lessc51_out_gg(s(T67), s(T68))

Pi is empty.
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:

LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)

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:

LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)

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:

  • LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)
    The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)

The TRS R consists of the following rules:

leqc33_in_gg(0, 0) → leqc33_out_gg(0, 0)
leqc33_in_gg(0, s(0)) → leqc33_out_gg(0, s(0))
leqc33_in_gg(s(T41), s(T42)) → U25_gg(T41, T42, leqc33_in_gg(T41, T42))
U25_gg(T41, T42, leqc33_out_gg(T41, T42)) → leqc33_out_gg(s(T41), s(T42))
lessc51_in_gg(0, s(0)) → lessc51_out_gg(0, s(0))
lessc51_in_gg(s(T67), s(T68)) → U26_gg(T67, T68, lessc51_in_gg(T67, T68))
U26_gg(T67, T68, lessc51_out_gg(T67, T68)) → lessc51_out_gg(s(T67), s(T68))

Pi is empty.
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:

LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)

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:

LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)

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:

  • LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)
    The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U9_GGA(T57, T58, T59, T60, T62, lessc51_in_gg(T59, T57))
U9_GGA(T57, T58, T59, T60, T62, lessc51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60, T62)
MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(0, T20), T22)
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → MERGE1_IN_GGA(.(s(0), T79), T81, T83)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U6_GGA(T35, T18, T36, T20, T22, leqc33_in_gg(T35, T36))
U6_GGA(T35, T18, T36, T20, T22, leqc33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20), T22)
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(s(0), T20), T22)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U13_GGA(T91, T79, T90, T81, T83, lessc51_in_gg(T90, T91))
U13_GGA(T91, T79, T90, T81, T83, lessc51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81, T83)

The TRS R consists of the following rules:

leqc33_in_gg(0, 0) → leqc33_out_gg(0, 0)
leqc33_in_gg(0, s(0)) → leqc33_out_gg(0, s(0))
leqc33_in_gg(s(T41), s(T42)) → U25_gg(T41, T42, leqc33_in_gg(T41, T42))
U25_gg(T41, T42, leqc33_out_gg(T41, T42)) → leqc33_out_gg(s(T41), s(T42))
lessc51_in_gg(0, s(0)) → lessc51_out_gg(0, s(0))
lessc51_in_gg(s(T67), s(T68)) → U26_gg(T67, T68, lessc51_in_gg(T67, T68))
U26_gg(T67, T68, lessc51_out_gg(T67, T68)) → lessc51_out_gg(s(T67), s(T68))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
leqc33_in_gg(x1, x2)  =  leqc33_in_gg(x1, x2)
leqc33_out_gg(x1, x2)  =  leqc33_out_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
lessc51_in_gg(x1, x2)  =  lessc51_in_gg(x1, x2)
lessc51_out_gg(x1, x2)  =  lessc51_out_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
MERGE1_IN_GGA(x1, x2, x3)  =  MERGE1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)
U9_GGA(x1, x2, x3, x4, x5, x6)  =  U9_GGA(x1, x2, x3, x4, x6)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)

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

(22) PiDPToQDPProof (SOUND transformation)

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

(23) Obligation:

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

MERGE1_IN_GGA(.(T57, T58), .(T59, T60)) → U9_GGA(T57, T58, T59, T60, lessc51_in_gg(T59, T57))
U9_GGA(T57, T58, T59, T60, lessc51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60)
MERGE1_IN_GGA(.(0, T18), .(0, T20)) → MERGE1_IN_GGA(T18, .(0, T20))
MERGE1_IN_GGA(.(s(0), T79), .(0, T81)) → MERGE1_IN_GGA(.(s(0), T79), T81)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20)) → U6_GGA(T35, T18, T36, T20, leqc33_in_gg(T35, T36))
U6_GGA(T35, T18, T36, T20, leqc33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20))
MERGE1_IN_GGA(.(0, T18), .(s(0), T20)) → MERGE1_IN_GGA(T18, .(s(0), T20))
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81)) → U13_GGA(T91, T79, T90, T81, lessc51_in_gg(T90, T91))
U13_GGA(T91, T79, T90, T81, lessc51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81)

The TRS R consists of the following rules:

leqc33_in_gg(0, 0) → leqc33_out_gg(0, 0)
leqc33_in_gg(0, s(0)) → leqc33_out_gg(0, s(0))
leqc33_in_gg(s(T41), s(T42)) → U25_gg(T41, T42, leqc33_in_gg(T41, T42))
U25_gg(T41, T42, leqc33_out_gg(T41, T42)) → leqc33_out_gg(s(T41), s(T42))
lessc51_in_gg(0, s(0)) → lessc51_out_gg(0, s(0))
lessc51_in_gg(s(T67), s(T68)) → U26_gg(T67, T68, lessc51_in_gg(T67, T68))
U26_gg(T67, T68, lessc51_out_gg(T67, T68)) → lessc51_out_gg(s(T67), s(T68))

The set Q consists of the following terms:

leqc33_in_gg(x0, x1)
U25_gg(x0, x1, x2)
lessc51_in_gg(x0, x1)
U26_gg(x0, x1, x2)

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

(24) 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:

MERGE1_IN_GGA(.(T57, T58), .(T59, T60)) → U9_GGA(T57, T58, T59, T60, lessc51_in_gg(T59, T57))
MERGE1_IN_GGA(.(0, T18), .(0, T20)) → MERGE1_IN_GGA(T18, .(0, T20))
MERGE1_IN_GGA(.(s(0), T79), .(0, T81)) → MERGE1_IN_GGA(.(s(0), T79), T81)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20)) → U6_GGA(T35, T18, T36, T20, leqc33_in_gg(T35, T36))
U6_GGA(T35, T18, T36, T20, leqc33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20))
MERGE1_IN_GGA(.(0, T18), .(s(0), T20)) → MERGE1_IN_GGA(T18, .(s(0), T20))
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81)) → U13_GGA(T91, T79, T90, T81, lessc51_in_gg(T90, T91))
U13_GGA(T91, T79, T90, T81, lessc51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81)


Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 2 + x1 + x2   
POL(0) = 0   
POL(MERGE1_IN_GGA(x1, x2)) = 2·x1 + 2·x2   
POL(U13_GGA(x1, x2, x3, x4, x5)) = 1 + 2·x1 + 2·x2 + 2·x3 + 2·x4 + 2·x5   
POL(U25_gg(x1, x2, x3)) = x1 + x2 + x3   
POL(U26_gg(x1, x2, x3)) = x1 + x2 + x3   
POL(U6_GGA(x1, x2, x3, x4, x5)) = 1 + 2·x1 + 2·x2 + 2·x3 + 2·x4 + 2·x5   
POL(U9_GGA(x1, x2, x3, x4, x5)) = 2 + x1 + 2·x2 + x3 + 2·x4 + x5   
POL(leqc33_in_gg(x1, x2)) = 2 + x1 + x2   
POL(leqc33_out_gg(x1, x2)) = 2 + x1 + x2   
POL(lessc51_in_gg(x1, x2)) = 2 + x1 + x2   
POL(lessc51_out_gg(x1, x2)) = 2 + x1 + x2   
POL(s(x1)) = 2·x1   

(25) Obligation:

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

U9_GGA(T57, T58, T59, T60, lessc51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60)

The TRS R consists of the following rules:

leqc33_in_gg(0, 0) → leqc33_out_gg(0, 0)
leqc33_in_gg(0, s(0)) → leqc33_out_gg(0, s(0))
leqc33_in_gg(s(T41), s(T42)) → U25_gg(T41, T42, leqc33_in_gg(T41, T42))
U25_gg(T41, T42, leqc33_out_gg(T41, T42)) → leqc33_out_gg(s(T41), s(T42))
lessc51_in_gg(0, s(0)) → lessc51_out_gg(0, s(0))
lessc51_in_gg(s(T67), s(T68)) → U26_gg(T67, T68, lessc51_in_gg(T67, T68))
U26_gg(T67, T68, lessc51_out_gg(T67, T68)) → lessc51_out_gg(s(T67), s(T68))

The set Q consists of the following terms:

leqc33_in_gg(x0, x1)
U25_gg(x0, x1, x2)
lessc51_in_gg(x0, x1)
U26_gg(x0, x1, x2)

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

(26) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(27) TRUE