Left Termination of the query pattern ms_in_2(a, g) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

ms([], []).
ms(.(X, []), .(X, [])).
ms(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(ms(X1s, Y1s), ','(ms(X2s, Y2s), merge(Y1s, Y2s, Ys)))).
split([], [], []).
split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys).
merge([], Xs, Xs).
merge(Xs, [], Xs).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(less(X, s(Y)), merge(Xs, .(Y, Ys), Zs)).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs)).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

ms(a,g).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MS_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
MS_IN(.(X, .(Y, Xs)), Ys) → SPLIT_IN(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → U51(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN(X1s, Y1s)
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U31(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → MS_IN(X2s, Y2s)
U31(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U41(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
U31(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → MERGE_IN(Y1s, Y2s, Ys)
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U81(X, Xs, Y, Ys, Zs, less_in(Y, X))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U101(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U91(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN(X, s(Y))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U71(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → MERGE_IN(Xs, .(Y, Ys), Zs)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
SPLIT_IN(x1, x2, x3)  =  SPLIT_IN
MERGE_IN(x1, x2, x3)  =  MERGE_IN
U21(x1, x2, x3, x4, x5, x6)  =  U21(x6)
U81(x1, x2, x3, x4, x5, x6)  =  U81(x6)
U11(x1, x2, x3, x4, x5)  =  U11(x5)
U31(x1, x2, x3, x4, x5, x6)  =  U31(x6)
U71(x1, x2, x3, x4, x5, x6)  =  U71(x6)
LESS_IN(x1, x2)  =  LESS_IN
U51(x1, x2, x3, x4, x5)  =  U51(x5)
U101(x1, x2, x3)  =  U101(x3)
MS_IN(x1, x2)  =  MS_IN
U41(x1, x2, x3, x4, x5)  =  U41(x5)
U91(x1, x2, x3, x4, x5, x6)  =  U91(x6)
U61(x1, x2, x3, x4, x5, x6)  =  U61(x6)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

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

MS_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
MS_IN(.(X, .(Y, Xs)), Ys) → SPLIT_IN(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → U51(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN(X1s, Y1s)
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U31(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → MS_IN(X2s, Y2s)
U31(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U41(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
U31(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → MERGE_IN(Y1s, Y2s, Ys)
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U81(X, Xs, Y, Ys, Zs, less_in(Y, X))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U101(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U91(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN(X, s(Y))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U71(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → MERGE_IN(Xs, .(Y, Ys), Zs)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
SPLIT_IN(x1, x2, x3)  =  SPLIT_IN
MERGE_IN(x1, x2, x3)  =  MERGE_IN
U21(x1, x2, x3, x4, x5, x6)  =  U21(x6)
U81(x1, x2, x3, x4, x5, x6)  =  U81(x6)
U11(x1, x2, x3, x4, x5)  =  U11(x5)
U31(x1, x2, x3, x4, x5, x6)  =  U31(x6)
U71(x1, x2, x3, x4, x5, x6)  =  U71(x6)
LESS_IN(x1, x2)  =  LESS_IN
U51(x1, x2, x3, x4, x5)  =  U51(x5)
U101(x1, x2, x3)  =  U101(x3)
MS_IN(x1, x2)  =  MS_IN
U41(x1, x2, x3, x4, x5)  =  U41(x5)
U91(x1, x2, x3, x4, x5, x6)  =  U91(x6)
U61(x1, x2, x3, x4, x5, x6)  =  U61(x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 10 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
LESS_IN(x1, x2)  =  LESS_IN

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_IN → LESS_IN

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LESS_IN → LESS_IN

The TRS R consists of the following rules:none


s = LESS_IN evaluates to t =LESS_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS_IN to LESS_IN.





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U81(X, Xs, Y, Ys, Zs, less_in(Y, X))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → MERGE_IN(Xs, .(Y, Ys), Zs)
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
MERGE_IN(x1, x2, x3)  =  MERGE_IN
U81(x1, x2, x3, x4, x5, x6)  =  U81(x6)
U61(x1, x2, x3, x4, x5, x6)  =  U61(x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U81(X, Xs, Y, Ys, Zs, less_in(Y, X))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → MERGE_IN(Xs, .(Y, Ys), Zs)
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
MERGE_IN(x1, x2, x3)  =  MERGE_IN
U81(x1, x2, x3, x4, x5, x6)  =  U81(x6)
U61(x1, x2, x3, x4, x5, x6)  =  U61(x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U61(less_out(X)) → MERGE_IN
U81(less_out(Y)) → MERGE_IN
MERGE_IN → U81(less_in)
MERGE_IN → U61(less_in)

The TRS R consists of the following rules:

less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))

The set Q consists of the following terms:

less_in
U10(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGE_IN → U61(less_in) at position [0] we obtained the following new rules:

MERGE_IN → U61(less_out(0))
MERGE_IN → U61(U10(less_in))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MERGE_IN → U61(less_out(0))
U61(less_out(X)) → MERGE_IN
MERGE_IN → U61(U10(less_in))
MERGE_IN → U81(less_in)
U81(less_out(Y)) → MERGE_IN

The TRS R consists of the following rules:

less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))

The set Q consists of the following terms:

less_in
U10(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGE_IN → U81(less_in) at position [0] we obtained the following new rules:

MERGE_IN → U81(less_out(0))
MERGE_IN → U81(U10(less_in))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U61(less_out(X)) → MERGE_IN
MERGE_IN → U61(less_out(0))
MERGE_IN → U61(U10(less_in))
MERGE_IN → U81(less_out(0))
MERGE_IN → U81(U10(less_in))
U81(less_out(Y)) → MERGE_IN

The TRS R consists of the following rules:

less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))

The set Q consists of the following terms:

less_in
U10(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U61(less_out(X)) → MERGE_IN
MERGE_IN → U61(less_out(0))
MERGE_IN → U61(U10(less_in))
MERGE_IN → U81(less_out(0))
MERGE_IN → U81(U10(less_in))
U81(less_out(Y)) → MERGE_IN

The TRS R consists of the following rules:

less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))


s = MERGE_IN evaluates to t =MERGE_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

MERGE_IN → U61(less_out(0))
with rule MERGE_IN → U61(less_out(0)) at position [] and matcher [ ]

U61(less_out(0)) → MERGE_IN
with rule U61(less_out(X)) → MERGE_IN

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
SPLIT_IN(x1, x2, x3)  =  SPLIT_IN

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

SPLIT_IN → SPLIT_IN

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

SPLIT_IN → SPLIT_IN

The TRS R consists of the following rules:none


s = SPLIT_IN evaluates to t =SPLIT_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SPLIT_IN to SPLIT_IN.





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN(X1s, Y1s)
MS_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → MS_IN(X2s, Y2s)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
U21(x1, x2, x3, x4, x5, x6)  =  U21(x6)
U11(x1, x2, x3, x4, x5)  =  U11(x5)
MS_IN(x1, x2)  =  MS_IN

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
                  ↳ QDP
                    ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U21(ms_out) → MS_IN
U11(split_out) → U21(ms_in)
MS_IN → U11(split_in)
U11(split_out) → MS_IN

The TRS R consists of the following rules:

ms_in → U1(split_in)
split_in → U5(split_in)
split_in → split_out
U5(split_out) → split_out
U1(split_out) → U2(ms_in)
ms_in → ms_out
U2(ms_out) → U3(ms_in)
U3(ms_out) → U4(merge_in)
merge_in → U8(less_in)
less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))
U8(less_out(Y)) → U9(merge_in)
merge_in → U6(less_in)
U6(less_out(X)) → U7(merge_in)
merge_in → merge_out
U7(merge_out) → merge_out
U9(merge_out) → merge_out
U4(merge_out) → ms_out

The set Q consists of the following terms:

ms_in
split_in
U5(x0)
U1(x0)
U2(x0)
U3(x0)
merge_in
less_in
U10(x0)
U8(x0)
U6(x0)
U7(x0)
U9(x0)
U4(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U11(split_out) → U21(ms_in) at position [0] we obtained the following new rules:

U11(split_out) → U21(U1(split_in))
U11(split_out) → U21(ms_out)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U11(split_out) → U21(U1(split_in))
U21(ms_out) → MS_IN
U11(split_out) → MS_IN
MS_IN → U11(split_in)
U11(split_out) → U21(ms_out)

The TRS R consists of the following rules:

ms_in → U1(split_in)
split_in → U5(split_in)
split_in → split_out
U5(split_out) → split_out
U1(split_out) → U2(ms_in)
ms_in → ms_out
U2(ms_out) → U3(ms_in)
U3(ms_out) → U4(merge_in)
merge_in → U8(less_in)
less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))
U8(less_out(Y)) → U9(merge_in)
merge_in → U6(less_in)
U6(less_out(X)) → U7(merge_in)
merge_in → merge_out
U7(merge_out) → merge_out
U9(merge_out) → merge_out
U4(merge_out) → ms_out

The set Q consists of the following terms:

ms_in
split_in
U5(x0)
U1(x0)
U2(x0)
U3(x0)
merge_in
less_in
U10(x0)
U8(x0)
U6(x0)
U7(x0)
U9(x0)
U4(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MS_IN → U11(split_in) at position [0] we obtained the following new rules:

MS_IN → U11(U5(split_in))
MS_IN → U11(split_out)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

U11(split_out) → U21(U1(split_in))
MS_IN → U11(split_out)
U21(ms_out) → MS_IN
U11(split_out) → MS_IN
MS_IN → U11(U5(split_in))
U11(split_out) → U21(ms_out)

The TRS R consists of the following rules:

ms_in → U1(split_in)
split_in → U5(split_in)
split_in → split_out
U5(split_out) → split_out
U1(split_out) → U2(ms_in)
ms_in → ms_out
U2(ms_out) → U3(ms_in)
U3(ms_out) → U4(merge_in)
merge_in → U8(less_in)
less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))
U8(less_out(Y)) → U9(merge_in)
merge_in → U6(less_in)
U6(less_out(X)) → U7(merge_in)
merge_in → merge_out
U7(merge_out) → merge_out
U9(merge_out) → merge_out
U4(merge_out) → ms_out

The set Q consists of the following terms:

ms_in
split_in
U5(x0)
U1(x0)
U2(x0)
U3(x0)
merge_in
less_in
U10(x0)
U8(x0)
U6(x0)
U7(x0)
U9(x0)
U4(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U11(split_out) → U21(U1(split_in))
MS_IN → U11(split_out)
U21(ms_out) → MS_IN
U11(split_out) → MS_IN
MS_IN → U11(U5(split_in))
U11(split_out) → U21(ms_out)

The TRS R consists of the following rules:

ms_in → U1(split_in)
split_in → U5(split_in)
split_in → split_out
U5(split_out) → split_out
U1(split_out) → U2(ms_in)
ms_in → ms_out
U2(ms_out) → U3(ms_in)
U3(ms_out) → U4(merge_in)
merge_in → U8(less_in)
less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))
U8(less_out(Y)) → U9(merge_in)
merge_in → U6(less_in)
U6(less_out(X)) → U7(merge_in)
merge_in → merge_out
U7(merge_out) → merge_out
U9(merge_out) → merge_out
U4(merge_out) → ms_out


s = U11(split_out) evaluates to t =U11(split_out)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

U11(split_out) → MS_IN
with rule U11(split_out) → MS_IN at position [] and matcher [ ]

MS_IN → U11(split_out)
with rule MS_IN → U11(split_out)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MS_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
MS_IN(.(X, .(Y, Xs)), Ys) → SPLIT_IN(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → U51(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN(X1s, Y1s)
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U31(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → MS_IN(X2s, Y2s)
U31(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U41(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
U31(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → MERGE_IN(Y1s, Y2s, Ys)
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U81(X, Xs, Y, Ys, Zs, less_in(Y, X))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U101(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U91(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN(X, s(Y))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U71(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → MERGE_IN(Xs, .(Y, Ys), Zs)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
SPLIT_IN(x1, x2, x3)  =  SPLIT_IN
MERGE_IN(x1, x2, x3)  =  MERGE_IN
U21(x1, x2, x3, x4, x5, x6)  =  U21(x6)
U81(x1, x2, x3, x4, x5, x6)  =  U81(x6)
U11(x1, x2, x3, x4, x5)  =  U11(x5)
U31(x1, x2, x3, x4, x5, x6)  =  U31(x6)
U71(x1, x2, x3, x4, x5, x6)  =  U71(x6)
LESS_IN(x1, x2)  =  LESS_IN
U51(x1, x2, x3, x4, x5)  =  U51(x5)
U101(x1, x2, x3)  =  U101(x3)
MS_IN(x1, x2)  =  MS_IN
U41(x1, x2, x3, x4, x5)  =  U41(x5)
U91(x1, x2, x3, x4, x5, x6)  =  U91(x6)
U61(x1, x2, x3, x4, x5, x6)  =  U61(x6)

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

MS_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
MS_IN(.(X, .(Y, Xs)), Ys) → SPLIT_IN(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → U51(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN(X1s, Y1s)
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U31(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → MS_IN(X2s, Y2s)
U31(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U41(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
U31(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → MERGE_IN(Y1s, Y2s, Ys)
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U81(X, Xs, Y, Ys, Zs, less_in(Y, X))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U101(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U91(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN(X, s(Y))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U71(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → MERGE_IN(Xs, .(Y, Ys), Zs)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
SPLIT_IN(x1, x2, x3)  =  SPLIT_IN
MERGE_IN(x1, x2, x3)  =  MERGE_IN
U21(x1, x2, x3, x4, x5, x6)  =  U21(x6)
U81(x1, x2, x3, x4, x5, x6)  =  U81(x6)
U11(x1, x2, x3, x4, x5)  =  U11(x5)
U31(x1, x2, x3, x4, x5, x6)  =  U31(x6)
U71(x1, x2, x3, x4, x5, x6)  =  U71(x6)
LESS_IN(x1, x2)  =  LESS_IN
U51(x1, x2, x3, x4, x5)  =  U51(x5)
U101(x1, x2, x3)  =  U101(x3)
MS_IN(x1, x2)  =  MS_IN
U41(x1, x2, x3, x4, x5)  =  U41(x5)
U91(x1, x2, x3, x4, x5, x6)  =  U91(x6)
U61(x1, x2, x3, x4, x5, x6)  =  U61(x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 10 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
LESS_IN(x1, x2)  =  LESS_IN

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN → LESS_IN

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LESS_IN → LESS_IN

The TRS R consists of the following rules:none


s = LESS_IN evaluates to t =LESS_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS_IN to LESS_IN.





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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

MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U81(X, Xs, Y, Ys, Zs, less_in(Y, X))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → MERGE_IN(Xs, .(Y, Ys), Zs)
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
MERGE_IN(x1, x2, x3)  =  MERGE_IN
U81(x1, x2, x3, x4, x5, x6)  =  U81(x6)
U61(x1, x2, x3, x4, x5, x6)  =  U61(x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U81(X, Xs, Y, Ys, Zs, less_in(Y, X))
U61(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → MERGE_IN(Xs, .(Y, Ys), Zs)
U81(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
MERGE_IN(x1, x2, x3)  =  MERGE_IN
U81(x1, x2, x3, x4, x5, x6)  =  U81(x6)
U61(x1, x2, x3, x4, x5, x6)  =  U61(x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
              ↳ PiDP
              ↳ PiDP

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

U61(less_out(X)) → MERGE_IN
U81(less_out(Y)) → MERGE_IN
MERGE_IN → U81(less_in)
MERGE_IN → U61(less_in)

The TRS R consists of the following rules:

less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))

The set Q consists of the following terms:

less_in
U10(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGE_IN → U61(less_in) at position [0] we obtained the following new rules:

MERGE_IN → U61(less_out(0))
MERGE_IN → U61(U10(less_in))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
              ↳ PiDP
              ↳ PiDP

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

MERGE_IN → U61(less_out(0))
U61(less_out(X)) → MERGE_IN
MERGE_IN → U61(U10(less_in))
MERGE_IN → U81(less_in)
U81(less_out(Y)) → MERGE_IN

The TRS R consists of the following rules:

less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))

The set Q consists of the following terms:

less_in
U10(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGE_IN → U81(less_in) at position [0] we obtained the following new rules:

MERGE_IN → U81(less_out(0))
MERGE_IN → U81(U10(less_in))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP

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

U61(less_out(X)) → MERGE_IN
MERGE_IN → U61(less_out(0))
MERGE_IN → U61(U10(less_in))
MERGE_IN → U81(less_out(0))
MERGE_IN → U81(U10(less_in))
U81(less_out(Y)) → MERGE_IN

The TRS R consists of the following rules:

less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))

The set Q consists of the following terms:

less_in
U10(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U61(less_out(X)) → MERGE_IN
MERGE_IN → U61(less_out(0))
MERGE_IN → U61(U10(less_in))
MERGE_IN → U81(less_out(0))
MERGE_IN → U81(U10(less_in))
U81(less_out(Y)) → MERGE_IN

The TRS R consists of the following rules:

less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))


s = MERGE_IN evaluates to t =MERGE_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

MERGE_IN → U61(less_out(0))
with rule MERGE_IN → U61(less_out(0)) at position [] and matcher [ ]

U61(less_out(0)) → MERGE_IN
with rule U61(less_out(X)) → MERGE_IN

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
SPLIT_IN(x1, x2, x3)  =  SPLIT_IN

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP

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

SPLIT_IN → SPLIT_IN

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

SPLIT_IN → SPLIT_IN

The TRS R consists of the following rules:none


s = SPLIT_IN evaluates to t =SPLIT_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SPLIT_IN to SPLIT_IN.





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof

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

U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN(X1s, Y1s)
MS_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
U21(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → MS_IN(X2s, Y2s)

The TRS R consists of the following rules:

ms_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, ms_in(X1s, Y1s))
ms_in(.(X, []), .(X, [])) → ms_out(.(X, []), .(X, []))
ms_in([], []) → ms_out([], [])
U2(X, Y, Xs, Ys, X2s, ms_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, ms_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, ms_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U10(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U10(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U8(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U9(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, less_in(X, s(Y)))
U6(X, Xs, Y, Ys, Zs, less_out(X, s(Y))) → U7(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U9(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → ms_out(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in(x1, x2)  =  ms_in
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x5)
split_in(x1, x2, x3)  =  split_in
U5(x1, x2, x3, x4, x5)  =  U5(x5)
[]  =  []
split_out(x1, x2, x3)  =  split_out
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
ms_out(x1, x2)  =  ms_out
U3(x1, x2, x3, x4, x5, x6)  =  U3(x6)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
merge_in(x1, x2, x3)  =  merge_in
U8(x1, x2, x3, x4, x5, x6)  =  U8(x6)
less_in(x1, x2)  =  less_in
s(x1)  =  s(x1)
U10(x1, x2, x3)  =  U10(x3)
0  =  0
less_out(x1, x2)  =  less_out(x1)
U9(x1, x2, x3, x4, x5, x6)  =  U9(x6)
U6(x1, x2, x3, x4, x5, x6)  =  U6(x6)
U7(x1, x2, x3, x4, x5, x6)  =  U7(x6)
merge_out(x1, x2, x3)  =  merge_out
U21(x1, x2, x3, x4, x5, x6)  =  U21(x6)
U11(x1, x2, x3, x4, x5)  =  U11(x5)
MS_IN(x1, x2)  =  MS_IN

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
                  ↳ QDP
                    ↳ Narrowing

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

U21(ms_out) → MS_IN
U11(split_out) → U21(ms_in)
MS_IN → U11(split_in)
U11(split_out) → MS_IN

The TRS R consists of the following rules:

ms_in → U1(split_in)
split_in → U5(split_in)
split_in → split_out
U5(split_out) → split_out
U1(split_out) → U2(ms_in)
ms_in → ms_out
U2(ms_out) → U3(ms_in)
U3(ms_out) → U4(merge_in)
merge_in → U8(less_in)
less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))
U8(less_out(Y)) → U9(merge_in)
merge_in → U6(less_in)
U6(less_out(X)) → U7(merge_in)
merge_in → merge_out
U7(merge_out) → merge_out
U9(merge_out) → merge_out
U4(merge_out) → ms_out

The set Q consists of the following terms:

ms_in
split_in
U5(x0)
U1(x0)
U2(x0)
U3(x0)
merge_in
less_in
U10(x0)
U8(x0)
U6(x0)
U7(x0)
U9(x0)
U4(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U11(split_out) → U21(ms_in) at position [0] we obtained the following new rules:

U11(split_out) → U21(U1(split_in))
U11(split_out) → U21(ms_out)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing

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

U11(split_out) → U21(U1(split_in))
U21(ms_out) → MS_IN
U11(split_out) → MS_IN
MS_IN → U11(split_in)
U11(split_out) → U21(ms_out)

The TRS R consists of the following rules:

ms_in → U1(split_in)
split_in → U5(split_in)
split_in → split_out
U5(split_out) → split_out
U1(split_out) → U2(ms_in)
ms_in → ms_out
U2(ms_out) → U3(ms_in)
U3(ms_out) → U4(merge_in)
merge_in → U8(less_in)
less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))
U8(less_out(Y)) → U9(merge_in)
merge_in → U6(less_in)
U6(less_out(X)) → U7(merge_in)
merge_in → merge_out
U7(merge_out) → merge_out
U9(merge_out) → merge_out
U4(merge_out) → ms_out

The set Q consists of the following terms:

ms_in
split_in
U5(x0)
U1(x0)
U2(x0)
U3(x0)
merge_in
less_in
U10(x0)
U8(x0)
U6(x0)
U7(x0)
U9(x0)
U4(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MS_IN → U11(split_in) at position [0] we obtained the following new rules:

MS_IN → U11(U5(split_in))
MS_IN → U11(split_out)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ NonTerminationProof

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

U11(split_out) → U21(U1(split_in))
MS_IN → U11(split_out)
U21(ms_out) → MS_IN
U11(split_out) → MS_IN
MS_IN → U11(U5(split_in))
U11(split_out) → U21(ms_out)

The TRS R consists of the following rules:

ms_in → U1(split_in)
split_in → U5(split_in)
split_in → split_out
U5(split_out) → split_out
U1(split_out) → U2(ms_in)
ms_in → ms_out
U2(ms_out) → U3(ms_in)
U3(ms_out) → U4(merge_in)
merge_in → U8(less_in)
less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))
U8(less_out(Y)) → U9(merge_in)
merge_in → U6(less_in)
U6(less_out(X)) → U7(merge_in)
merge_in → merge_out
U7(merge_out) → merge_out
U9(merge_out) → merge_out
U4(merge_out) → ms_out

The set Q consists of the following terms:

ms_in
split_in
U5(x0)
U1(x0)
U2(x0)
U3(x0)
merge_in
less_in
U10(x0)
U8(x0)
U6(x0)
U7(x0)
U9(x0)
U4(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U11(split_out) → U21(U1(split_in))
MS_IN → U11(split_out)
U21(ms_out) → MS_IN
U11(split_out) → MS_IN
MS_IN → U11(U5(split_in))
U11(split_out) → U21(ms_out)

The TRS R consists of the following rules:

ms_in → U1(split_in)
split_in → U5(split_in)
split_in → split_out
U5(split_out) → split_out
U1(split_out) → U2(ms_in)
ms_in → ms_out
U2(ms_out) → U3(ms_in)
U3(ms_out) → U4(merge_in)
merge_in → U8(less_in)
less_in → U10(less_in)
less_in → less_out(0)
U10(less_out(X)) → less_out(s(X))
U8(less_out(Y)) → U9(merge_in)
merge_in → U6(less_in)
U6(less_out(X)) → U7(merge_in)
merge_in → merge_out
U7(merge_out) → merge_out
U9(merge_out) → merge_out
U4(merge_out) → ms_out


s = U11(split_out) evaluates to t =U11(split_out)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

U11(split_out) → MS_IN
with rule U11(split_out) → MS_IN at position [] and matcher [ ]

MS_IN → U11(split_out)
with rule MS_IN → U11(split_out)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.