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

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)

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_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)


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_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MS_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
MS_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MS_IN_AA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_AA(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → MERGE_IN_AAA(Y1s, Y2s, Ys)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN_AA(X, s(Y))
LESS_IN_AA(s(X), s(Y)) → U10_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → MERGE_IN_AAA(Xs, .(Y, Ys), Zs)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_AA(Y, X)
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → MERGE_IN_AAG(Y1s, Y2s, Ys)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN_GA(X, s(Y))
LESS_IN_GA(s(X), s(Y)) → U10_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → MERGE_IN_AAG(Xs, .(Y, Ys), Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GA(Y, X)
U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)
MERGE_IN_AAA(x1, x2, x3)  =  MERGE_IN_AAA
U10_GA(x1, x2, x3)  =  U10_GA(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)
U10_AA(x1, x2, x3)  =  U10_AA(x3)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_AG(x1, x2, x3, x4, x5, x6)  =  U3_AG(x4, x6)
U7_AAG(x1, x2, x3, x4, x5, x6)  =  U7_AAG(x1, x6)
MS_IN_AG(x1, x2)  =  MS_IN_AG(x2)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x4, x6)
U8_AAG(x1, x2, x3, x4, x5, x6)  =  U8_AAG(x3, x5, x6)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
U4_AA(x1, x2, x3, x4, x5)  =  U4_AA(x5)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(x3)
U9_AAG(x1, x2, x3, x4, x5, x6)  =  U9_AAG(x3, x6)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_AAA
U9_AAA(x1, x2, x3, x4, x5, x6)  =  U9_AAA(x6)
U7_AAA(x1, x2, x3, x4, x5, x6)  =  U7_AAA(x6)
U3_AA(x1, x2, x3, x4, x5, x6)  =  U3_AA(x6)
MS_IN_AA(x1, x2)  =  MS_IN_AA

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_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MS_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
MS_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MS_IN_AA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_AA(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → MERGE_IN_AAA(Y1s, Y2s, Ys)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN_AA(X, s(Y))
LESS_IN_AA(s(X), s(Y)) → U10_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → MERGE_IN_AAA(Xs, .(Y, Ys), Zs)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_AA(Y, X)
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → MERGE_IN_AAG(Y1s, Y2s, Ys)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN_GA(X, s(Y))
LESS_IN_GA(s(X), s(Y)) → U10_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → MERGE_IN_AAG(Xs, .(Y, Ys), Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GA(Y, X)
U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)
MERGE_IN_AAA(x1, x2, x3)  =  MERGE_IN_AAA
U10_GA(x1, x2, x3)  =  U10_GA(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)
U10_AA(x1, x2, x3)  =  U10_AA(x3)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_AG(x1, x2, x3, x4, x5, x6)  =  U3_AG(x4, x6)
U7_AAG(x1, x2, x3, x4, x5, x6)  =  U7_AAG(x1, x6)
MS_IN_AG(x1, x2)  =  MS_IN_AG(x2)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x4, x6)
U8_AAG(x1, x2, x3, x4, x5, x6)  =  U8_AAG(x3, x5, x6)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
U4_AA(x1, x2, x3, x4, x5)  =  U4_AA(x5)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(x3)
U9_AAG(x1, x2, x3, x4, x5, x6)  =  U9_AAG(x3, x6)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_AAA
U9_AAA(x1, x2, x3, x4, x5, x6)  =  U9_AAA(x6)
U7_AAA(x1, x2, x3, x4, x5, x6)  =  U7_AAA(x6)
U3_AA(x1, x2, x3, x4, x5, x6)  =  U3_AA(x6)
MS_IN_AA(x1, x2)  =  MS_IN_AA

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 6 SCCs with 23 less nodes.

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

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

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

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
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



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

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

U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → MERGE_IN_AAG(Xs, .(Y, Ys), Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
U8_AAG(x1, x2, x3, x4, x5, x6)  =  U8_AAG(x3, x5, x6)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(x3)

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
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → MERGE_IN_AAG(Xs, .(Y, Ys), Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))

The TRS R consists of the following rules:

less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
U8_AAG(x1, x2, x3, x4, x5, x6)  =  U8_AAG(x3, x5, x6)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(x3)

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
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MERGE_IN_AAG(.(Y, Zs)) → U8_AAG(Y, Zs, less_in_ga(Y))
U8_AAG(Y, Zs, less_out_ga) → MERGE_IN_AAG(Zs)
MERGE_IN_AAG(.(X, Zs)) → U6_AAG(X, Zs, less_in_ga(X))
U6_AAG(X, Zs, less_out_ga) → MERGE_IN_AAG(Zs)

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U10_ga(less_in_ga(X))
U10_ga(less_out_ga) → less_out_ga

The set Q consists of the following terms:

less_in_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA

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
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_IN_AA → LESS_IN_AA

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_AA → LESS_IN_AA

The TRS R consists of the following rules:none


s = LESS_IN_AA evaluates to t =LESS_IN_AA

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_AA to LESS_IN_AA.





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

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

U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → MERGE_IN_AAA(Xs, .(Y, Ys), Zs)
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)
MERGE_IN_AAA(x1, x2, x3)  =  MERGE_IN_AAA
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → MERGE_IN_AAA(Xs, .(Y, Ys), Zs)
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))

The TRS R consists of the following rules:

less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
0  =  0
s(x1)  =  s(x1)
MERGE_IN_AAA(x1, x2, x3)  =  MERGE_IN_AAA
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U8_AAA(less_out_aa(Y)) → MERGE_IN_AAA
MERGE_IN_AAA → U8_AAA(less_in_aa)
U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(less_in_aa)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

less_in_aa
U10_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGE_IN_AAA → U8_AAA(less_in_aa) at position [0] we obtained the following new rules:

MERGE_IN_AAA → U8_AAA(U10_aa(less_in_aa))
MERGE_IN_AAA → U8_AAA(less_out_aa(0))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ 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_AAA → U8_AAA(U10_aa(less_in_aa))
U8_AAA(less_out_aa(Y)) → MERGE_IN_AAA
U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(less_in_aa)
MERGE_IN_AAA → U8_AAA(less_out_aa(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

less_in_aa
U10_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGE_IN_AAA → U6_AAA(less_in_aa) at position [0] we obtained the following new rules:

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
MERGE_IN_AAA → U6_AAA(U10_aa(less_in_aa))



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

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
MERGE_IN_AAA → U8_AAA(U10_aa(less_in_aa))
U8_AAA(less_out_aa(Y)) → MERGE_IN_AAA
U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(U10_aa(less_in_aa))
MERGE_IN_AAA → U8_AAA(less_out_aa(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

less_in_aa
U10_aa(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:

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
MERGE_IN_AAA → U8_AAA(U10_aa(less_in_aa))
U8_AAA(less_out_aa(Y)) → MERGE_IN_AAA
U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(U10_aa(less_in_aa))
MERGE_IN_AAA → U8_AAA(less_out_aa(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U10_aa(less_out_aa(X)) → less_out_aa(s(X))


s = U6_AAA(less_out_aa(X)) evaluates to t =U6_AAA(less_out_aa(0))

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



Rewriting sequence

U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
with rule U6_AAA(less_out_aa(X')) → MERGE_IN_AAA at position [] and matcher [X' / X]

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
with rule MERGE_IN_AAA → U6_AAA(less_out_aa(0))

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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_AAA

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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

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

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

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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

SPLIT_IN_AAA → SPLIT_IN_AAA

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_AAA → SPLIT_IN_AAA

The TRS R consists of the following rules:none


s = SPLIT_IN_AAA evaluates to t =SPLIT_IN_AAA

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_AAA to SPLIT_IN_AAA.





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

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

U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
MS_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x6)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
MS_IN_AA(x1, x2)  =  MS_IN_AA

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
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
MS_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))

The TRS R consists of the following rules:

ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
0  =  0
s(x1)  =  s(x1)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
MS_IN_AA(x1, x2)  =  MS_IN_AA

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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U2_AA(ms_out_aa) → MS_IN_AA
MS_IN_AA → U1_AA(split_in_aaa)
U1_AA(split_out_aaa) → U2_AA(ms_in_aa)
U1_AA(split_out_aaa) → MS_IN_AA

The TRS R consists of the following rules:

ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
split_in_aaa → U5_aaa(split_in_aaa)
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
U5_aaa(split_out_aaa) → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U3_aa(ms_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → ms_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(less_in_aa)
merge_in_aaa → U8_aaa(less_in_aa)
U6_aaa(less_out_aa(X)) → U7_aaa(merge_in_aaa)
U8_aaa(less_out_aa(Y)) → U9_aaa(merge_in_aaa)
less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U7_aaa(merge_out_aaa) → merge_out_aaa
U9_aaa(merge_out_aaa) → merge_out_aaa
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

ms_in_aa
split_in_aaa
U1_aa(x0)
U5_aaa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
U8_aaa(x0)
less_in_aa
U7_aaa(x0)
U9_aaa(x0)
U10_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_AA(split_out_aaa) → U2_AA(ms_in_aa) at position [0] we obtained the following new rules:

U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(ms_out_aa)



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

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

U2_AA(ms_out_aa) → MS_IN_AA
MS_IN_AA → U1_AA(split_in_aaa)
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(ms_out_aa)
U1_AA(split_out_aaa) → MS_IN_AA

The TRS R consists of the following rules:

ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
split_in_aaa → U5_aaa(split_in_aaa)
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
U5_aaa(split_out_aaa) → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U3_aa(ms_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → ms_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(less_in_aa)
merge_in_aaa → U8_aaa(less_in_aa)
U6_aaa(less_out_aa(X)) → U7_aaa(merge_in_aaa)
U8_aaa(less_out_aa(Y)) → U9_aaa(merge_in_aaa)
less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U7_aaa(merge_out_aaa) → merge_out_aaa
U9_aaa(merge_out_aaa) → merge_out_aaa
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

ms_in_aa
split_in_aaa
U1_aa(x0)
U5_aaa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
U8_aaa(x0)
less_in_aa
U7_aaa(x0)
U9_aaa(x0)
U10_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MS_IN_AA → U1_AA(split_in_aaa) at position [0] we obtained the following new rules:

MS_IN_AA → U1_AA(split_out_aaa)
MS_IN_AA → U1_AA(U5_aaa(split_in_aaa))



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

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

U2_AA(ms_out_aa) → MS_IN_AA
MS_IN_AA → U1_AA(U5_aaa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
MS_IN_AA → U1_AA(split_out_aaa)
U1_AA(split_out_aaa) → MS_IN_AA
U1_AA(split_out_aaa) → U2_AA(ms_out_aa)

The TRS R consists of the following rules:

ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
split_in_aaa → U5_aaa(split_in_aaa)
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
U5_aaa(split_out_aaa) → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U3_aa(ms_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → ms_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(less_in_aa)
merge_in_aaa → U8_aaa(less_in_aa)
U6_aaa(less_out_aa(X)) → U7_aaa(merge_in_aaa)
U8_aaa(less_out_aa(Y)) → U9_aaa(merge_in_aaa)
less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U7_aaa(merge_out_aaa) → merge_out_aaa
U9_aaa(merge_out_aaa) → merge_out_aaa
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

ms_in_aa
split_in_aaa
U1_aa(x0)
U5_aaa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
U8_aaa(x0)
less_in_aa
U7_aaa(x0)
U9_aaa(x0)
U10_aa(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:

U2_AA(ms_out_aa) → MS_IN_AA
MS_IN_AA → U1_AA(U5_aaa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
MS_IN_AA → U1_AA(split_out_aaa)
U1_AA(split_out_aaa) → MS_IN_AA
U1_AA(split_out_aaa) → U2_AA(ms_out_aa)

The TRS R consists of the following rules:

ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
split_in_aaa → U5_aaa(split_in_aaa)
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
U5_aaa(split_out_aaa) → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U3_aa(ms_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → ms_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(less_in_aa)
merge_in_aaa → U8_aaa(less_in_aa)
U6_aaa(less_out_aa(X)) → U7_aaa(merge_in_aaa)
U8_aaa(less_out_aa(Y)) → U9_aaa(merge_in_aaa)
less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U7_aaa(merge_out_aaa) → merge_out_aaa
U9_aaa(merge_out_aaa) → merge_out_aaa
U10_aa(less_out_aa(X)) → less_out_aa(s(X))


s = U1_AA(split_out_aaa) evaluates to t =U1_AA(split_out_aaa)

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



Rewriting sequence

U1_AA(split_out_aaa) → MS_IN_AA
with rule U1_AA(split_out_aaa) → MS_IN_AA at position [] and matcher [ ]

MS_IN_AA → U1_AA(split_out_aaa)
with rule MS_IN_AA → U1_AA(split_out_aaa)

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

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)

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_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)


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_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MS_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
MS_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MS_IN_AA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_AA(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → MERGE_IN_AAA(Y1s, Y2s, Ys)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN_AA(X, s(Y))
LESS_IN_AA(s(X), s(Y)) → U10_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → MERGE_IN_AAA(Xs, .(Y, Ys), Zs)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_AA(Y, X)
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → MERGE_IN_AAG(Y1s, Y2s, Ys)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN_GA(X, s(Y))
LESS_IN_GA(s(X), s(Y)) → U10_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → MERGE_IN_AAG(Xs, .(Y, Ys), Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GA(Y, X)
U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)
MERGE_IN_AAA(x1, x2, x3)  =  MERGE_IN_AAA
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)
U10_AA(x1, x2, x3)  =  U10_AA(x3)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_AG(x1, x2, x3, x4, x5, x6)  =  U3_AG(x4, x6)
U7_AAG(x1, x2, x3, x4, x5, x6)  =  U7_AAG(x1, x5, x6)
MS_IN_AG(x1, x2)  =  MS_IN_AG(x2)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x4, x5)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x4, x6)
U8_AAG(x1, x2, x3, x4, x5, x6)  =  U8_AAG(x3, x5, x6)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
U4_AA(x1, x2, x3, x4, x5)  =  U4_AA(x5)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(x3)
U9_AAG(x1, x2, x3, x4, x5, x6)  =  U9_AAG(x3, x5, x6)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_AAA
U9_AAA(x1, x2, x3, x4, x5, x6)  =  U9_AAA(x6)
U7_AAA(x1, x2, x3, x4, x5, x6)  =  U7_AAA(x6)
U3_AA(x1, x2, x3, x4, x5, x6)  =  U3_AA(x6)
MS_IN_AA(x1, x2)  =  MS_IN_AA

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_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MS_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
MS_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MS_IN_AA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_AA(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → MERGE_IN_AAA(Y1s, Y2s, Ys)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN_AA(X, s(Y))
LESS_IN_AA(s(X), s(Y)) → U10_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → MERGE_IN_AAA(Xs, .(Y, Ys), Zs)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_AA(Y, X)
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → MERGE_IN_AAG(Y1s, Y2s, Ys)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → LESS_IN_GA(X, s(Y))
LESS_IN_GA(s(X), s(Y)) → U10_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → MERGE_IN_AAG(Xs, .(Y, Ys), Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GA(Y, X)
U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)
MERGE_IN_AAA(x1, x2, x3)  =  MERGE_IN_AAA
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)
U10_AA(x1, x2, x3)  =  U10_AA(x3)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_AG(x1, x2, x3, x4, x5, x6)  =  U3_AG(x4, x6)
U7_AAG(x1, x2, x3, x4, x5, x6)  =  U7_AAG(x1, x5, x6)
MS_IN_AG(x1, x2)  =  MS_IN_AG(x2)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x4, x5)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
U2_AG(x1, x2, x3, x4, x5, x6)  =  U2_AG(x4, x6)
U8_AAG(x1, x2, x3, x4, x5, x6)  =  U8_AAG(x3, x5, x6)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
U4_AA(x1, x2, x3, x4, x5)  =  U4_AA(x5)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(x3)
U9_AAG(x1, x2, x3, x4, x5, x6)  =  U9_AAG(x3, x5, x6)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_AAA
U9_AAA(x1, x2, x3, x4, x5, x6)  =  U9_AAA(x6)
U7_AAA(x1, x2, x3, x4, x5, x6)  =  U7_AAA(x6)
U3_AA(x1, x2, x3, x4, x5, x6)  =  U3_AA(x6)
MS_IN_AA(x1, x2)  =  MS_IN_AA

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 6 SCCs with 23 less nodes.

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

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

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

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
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



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

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

U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → MERGE_IN_AAG(Xs, .(Y, Ys), Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
U8_AAG(x1, x2, x3, x4, x5, x6)  =  U8_AAG(x3, x5, x6)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(x3)

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
              ↳ PiDP
              ↳ PiDP

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

U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → MERGE_IN_AAG(Xs, .(Y, Ys), Zs)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))

The TRS R consists of the following rules:

less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
U8_AAG(x1, x2, x3, x4, x5, x6)  =  U8_AAG(x3, x5, x6)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(x3)

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
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

MERGE_IN_AAG(.(Y, Zs)) → U8_AAG(Y, Zs, less_in_ga(Y))
MERGE_IN_AAG(.(X, Zs)) → U6_AAG(X, Zs, less_in_ga(X))
U6_AAG(X, Zs, less_out_ga(X)) → MERGE_IN_AAG(Zs)
U8_AAG(Y, Zs, less_out_ga(Y)) → MERGE_IN_AAG(Zs)

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U10_ga(X, less_in_ga(X))
U10_ga(X, less_out_ga(X)) → less_out_ga(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA

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
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_AA → LESS_IN_AA

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_AA → LESS_IN_AA

The TRS R consists of the following rules:none


s = LESS_IN_AA evaluates to t =LESS_IN_AA

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_AA to LESS_IN_AA.





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

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

U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → MERGE_IN_AAA(Xs, .(Y, Ys), Zs)
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)
MERGE_IN_AAA(x1, x2, x3)  =  MERGE_IN_AAA
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → MERGE_IN_AAA(Xs, .(Y, Ys), Zs)
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))

The TRS R consists of the following rules:

less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
0  =  0
s(x1)  =  s(x1)
MERGE_IN_AAA(x1, x2, x3)  =  MERGE_IN_AAA
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
              ↳ PiDP
              ↳ PiDP

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

U8_AAA(less_out_aa(Y)) → MERGE_IN_AAA
MERGE_IN_AAA → U8_AAA(less_in_aa)
U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(less_in_aa)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

less_in_aa
U10_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGE_IN_AAA → U8_AAA(less_in_aa) at position [0] we obtained the following new rules:

MERGE_IN_AAA → U8_AAA(U10_aa(less_in_aa))
MERGE_IN_AAA → U8_AAA(less_out_aa(0))



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

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

MERGE_IN_AAA → U8_AAA(U10_aa(less_in_aa))
U8_AAA(less_out_aa(Y)) → MERGE_IN_AAA
U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(less_in_aa)
MERGE_IN_AAA → U8_AAA(less_out_aa(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

less_in_aa
U10_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGE_IN_AAA → U6_AAA(less_in_aa) at position [0] we obtained the following new rules:

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
MERGE_IN_AAA → U6_AAA(U10_aa(less_in_aa))



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

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

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
MERGE_IN_AAA → U8_AAA(U10_aa(less_in_aa))
U8_AAA(less_out_aa(Y)) → MERGE_IN_AAA
U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(U10_aa(less_in_aa))
MERGE_IN_AAA → U8_AAA(less_out_aa(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

less_in_aa
U10_aa(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:

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
MERGE_IN_AAA → U8_AAA(U10_aa(less_in_aa))
U8_AAA(less_out_aa(Y)) → MERGE_IN_AAA
U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(U10_aa(less_in_aa))
MERGE_IN_AAA → U8_AAA(less_out_aa(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U10_aa(less_out_aa(X)) → less_out_aa(s(X))


s = U6_AAA(less_out_aa(X)) evaluates to t =U6_AAA(less_out_aa(0))

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



Rewriting sequence

U6_AAA(less_out_aa(X)) → MERGE_IN_AAA
with rule U6_AAA(less_out_aa(X')) → MERGE_IN_AAA at position [] and matcher [X' / X]

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
with rule MERGE_IN_AAA → U6_AAA(less_out_aa(0))

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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

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

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_AAA

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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

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

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

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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP

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

SPLIT_IN_AAA → SPLIT_IN_AAA

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_AAA → SPLIT_IN_AAA

The TRS R consists of the following rules:none


s = SPLIT_IN_AAA evaluates to t =SPLIT_IN_AAA

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_AAA to SPLIT_IN_AAA.





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

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

U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
MS_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))

The TRS R consists of the following rules:

ms_in_ag([], []) → ms_out_ag([], [])
ms_in_ag(.(X, []), .(X, [])) → ms_out_ag(.(X, []), .(X, []))
ms_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y)))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U10_ga(X, Y, less_in_ga(X, Y))
U10_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs))
merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X))
U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) → U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → ms_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
ms_in_ag(x1, x2)  =  ms_in_ag(x2)
[]  =  []
ms_out_ag(x1, x2)  =  ms_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6)  =  U2_ag(x4, x6)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6)  =  U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x4, x5)
merge_in_aag(x1, x2, x3)  =  merge_in_aag(x3)
merge_out_aag(x1, x2, x3)  =  merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6)  =  U6_aag(x1, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U7_aag(x1, x2, x3, x4, x5, x6)  =  U7_aag(x1, x5, x6)
U8_aag(x1, x2, x3, x4, x5, x6)  =  U8_aag(x3, x5, x6)
U9_aag(x1, x2, x3, x4, x5, x6)  =  U9_aag(x3, x5, x6)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
MS_IN_AA(x1, x2)  =  MS_IN_AA

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
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → MS_IN_AA(X2s, Y2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MS_IN_AA(X1s, Y1s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
MS_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))

The TRS R consists of the following rules:

ms_in_aa([], []) → ms_out_aa([], [])
ms_in_aa(.(X, []), .(X, [])) → ms_out_aa(.(X, []), .(X, []))
ms_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → ms_out_aa(.(X, .(Y, Xs)), Ys)
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y)))
merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X))
U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs))
U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U10_aa(X, Y, less_in_aa(X, Y))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U10_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
split_in_aaa(x1, x2, x3)  =  split_in_aaa
split_out_aaa(x1, x2, x3)  =  split_out_aaa
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
ms_in_aa(x1, x2)  =  ms_in_aa
ms_out_aa(x1, x2)  =  ms_out_aa
U1_aa(x1, x2, x3, x4, x5)  =  U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6)  =  U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6)  =  U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5)  =  U4_aa(x5)
merge_in_aaa(x1, x2, x3)  =  merge_in_aaa
merge_out_aaa(x1, x2, x3)  =  merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U10_aa(x1, x2, x3)  =  U10_aa(x3)
U7_aaa(x1, x2, x3, x4, x5, x6)  =  U7_aaa(x6)
U8_aaa(x1, x2, x3, x4, x5, x6)  =  U8_aaa(x6)
U9_aaa(x1, x2, x3, x4, x5, x6)  =  U9_aaa(x6)
0  =  0
s(x1)  =  s(x1)
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)
MS_IN_AA(x1, x2)  =  MS_IN_AA

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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing

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

U2_AA(ms_out_aa) → MS_IN_AA
MS_IN_AA → U1_AA(split_in_aaa)
U1_AA(split_out_aaa) → U2_AA(ms_in_aa)
U1_AA(split_out_aaa) → MS_IN_AA

The TRS R consists of the following rules:

ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
split_in_aaa → U5_aaa(split_in_aaa)
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
U5_aaa(split_out_aaa) → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U3_aa(ms_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → ms_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(less_in_aa)
merge_in_aaa → U8_aaa(less_in_aa)
U6_aaa(less_out_aa(X)) → U7_aaa(merge_in_aaa)
U8_aaa(less_out_aa(Y)) → U9_aaa(merge_in_aaa)
less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U7_aaa(merge_out_aaa) → merge_out_aaa
U9_aaa(merge_out_aaa) → merge_out_aaa
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

ms_in_aa
split_in_aaa
U1_aa(x0)
U5_aaa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
U8_aaa(x0)
less_in_aa
U7_aaa(x0)
U9_aaa(x0)
U10_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_AA(split_out_aaa) → U2_AA(ms_in_aa) at position [0] we obtained the following new rules:

U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(ms_out_aa)



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

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

U2_AA(ms_out_aa) → MS_IN_AA
MS_IN_AA → U1_AA(split_in_aaa)
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(ms_out_aa)
U1_AA(split_out_aaa) → MS_IN_AA

The TRS R consists of the following rules:

ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
split_in_aaa → U5_aaa(split_in_aaa)
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
U5_aaa(split_out_aaa) → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U3_aa(ms_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → ms_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(less_in_aa)
merge_in_aaa → U8_aaa(less_in_aa)
U6_aaa(less_out_aa(X)) → U7_aaa(merge_in_aaa)
U8_aaa(less_out_aa(Y)) → U9_aaa(merge_in_aaa)
less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U7_aaa(merge_out_aaa) → merge_out_aaa
U9_aaa(merge_out_aaa) → merge_out_aaa
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

ms_in_aa
split_in_aaa
U1_aa(x0)
U5_aaa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
U8_aaa(x0)
less_in_aa
U7_aaa(x0)
U9_aaa(x0)
U10_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MS_IN_AA → U1_AA(split_in_aaa) at position [0] we obtained the following new rules:

MS_IN_AA → U1_AA(split_out_aaa)
MS_IN_AA → U1_AA(U5_aaa(split_in_aaa))



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

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

U2_AA(ms_out_aa) → MS_IN_AA
MS_IN_AA → U1_AA(U5_aaa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
MS_IN_AA → U1_AA(split_out_aaa)
U1_AA(split_out_aaa) → MS_IN_AA
U1_AA(split_out_aaa) → U2_AA(ms_out_aa)

The TRS R consists of the following rules:

ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
split_in_aaa → U5_aaa(split_in_aaa)
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
U5_aaa(split_out_aaa) → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U3_aa(ms_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → ms_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(less_in_aa)
merge_in_aaa → U8_aaa(less_in_aa)
U6_aaa(less_out_aa(X)) → U7_aaa(merge_in_aaa)
U8_aaa(less_out_aa(Y)) → U9_aaa(merge_in_aaa)
less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U7_aaa(merge_out_aaa) → merge_out_aaa
U9_aaa(merge_out_aaa) → merge_out_aaa
U10_aa(less_out_aa(X)) → less_out_aa(s(X))

The set Q consists of the following terms:

ms_in_aa
split_in_aaa
U1_aa(x0)
U5_aaa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
U8_aaa(x0)
less_in_aa
U7_aaa(x0)
U9_aaa(x0)
U10_aa(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:

U2_AA(ms_out_aa) → MS_IN_AA
MS_IN_AA → U1_AA(U5_aaa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
MS_IN_AA → U1_AA(split_out_aaa)
U1_AA(split_out_aaa) → MS_IN_AA
U1_AA(split_out_aaa) → U2_AA(ms_out_aa)

The TRS R consists of the following rules:

ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
split_in_aaa → U5_aaa(split_in_aaa)
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
U5_aaa(split_out_aaa) → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U3_aa(ms_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → ms_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(less_in_aa)
merge_in_aaa → U8_aaa(less_in_aa)
U6_aaa(less_out_aa(X)) → U7_aaa(merge_in_aaa)
U8_aaa(less_out_aa(Y)) → U9_aaa(merge_in_aaa)
less_in_aa → less_out_aa(0)
less_in_aa → U10_aa(less_in_aa)
U7_aaa(merge_out_aaa) → merge_out_aaa
U9_aaa(merge_out_aaa) → merge_out_aaa
U10_aa(less_out_aa(X)) → less_out_aa(s(X))


s = U1_AA(split_out_aaa) evaluates to t =U1_AA(split_out_aaa)

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



Rewriting sequence

U1_AA(split_out_aaa) → MS_IN_AA
with rule U1_AA(split_out_aaa) → MS_IN_AA at position [] and matcher [ ]

MS_IN_AA → U1_AA(split_out_aaa)
with rule MS_IN_AA → U1_AA(split_out_aaa)

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.