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

(0) Obligation:

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(X1)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

ms(a,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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

(2) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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)
MS_IN_AG(x1, x2)  =  MS_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_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
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
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
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)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(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)

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

(4) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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)
MS_IN_AG(x1, x2)  =  MS_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_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
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
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
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)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(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)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 23 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LESS_IN_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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS_IN_GA(s(X)) → LESS_IN_GA(X)
  The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

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)))
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)) → 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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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_AAG(x1, x2, x3)  =  MERGE_IN_AAG(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)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

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

The TRS R consists of the following rules:

less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
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)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(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)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• MERGE_IN_AAG(.(X, Zs)) → U6_AAG(X, Zs, less_in_ga(X))
  The graph contains the following edges 1 > 1, 1 > 2

• MERGE_IN_AAG(.(Y, Zs)) → U8_AAG(Y, Zs, less_in_ga(Y))
  The graph contains the following edges 1 > 1, 1 > 2

• U6_AAG(X, Zs, less_out_ga(X)) → MERGE_IN_AAG(Zs)
  The graph contains the following edges 2 >= 1

• U8_AAG(Y, Zs, less_out_ga(Y)) → MERGE_IN_AAG(Zs)
  The graph contains the following edges 2 >= 1

(20) TRUE

(21) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

LESS_IN_AA → LESS_IN_AA

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

(26) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

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

(27) FALSE

(28) Obligation:

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)
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))
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

The TRS R consists of the following rules:

less_in_aa(0, s(X1)) → less_out_aa(0, s(X1))
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
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

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

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.

(33) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE_IN_AAA → U6_AAA(less_in_aa) at position [0] we obtained the following new rules [LPAR04]:

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

(34) Obligation:

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

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

(35) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE_IN_AAA → U8_AAA(less_in_aa) at position [0] we obtained the following new rules [LPAR04]:

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

(36) Obligation:

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

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

(37) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = MERGE_IN_AAA evaluates to t =MERGE_IN_AAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
with rule MERGE_IN_AAA → U6_AAA(less_out_aa(0)) at position [] and matcher [ ]

U6_AAA(less_out_aa(0)) → MERGE_IN_AAA
with rule U6_AAA(less_out_aa(X)) → MERGE_IN_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.

(38) FALSE

(39) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) Obligation:

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

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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) Obligation:

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

SPLIT_IN_AAA → SPLIT_IN_AAA

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

(44) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

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

(45) FALSE

(46) Obligation:

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

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

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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)
MS_IN_AA(x1, x2)  =  MS_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)

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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) Obligation:

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

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

The TRS R consists of the following rules:

split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
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))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
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))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
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))
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(X1)) → less_out_aa(0, s(X1))
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)
MS_IN_AA(x1, x2)  =  MS_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)

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

(49) PiDPToQDPProof (SOUND transformation)

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

(50) Obligation:

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

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

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
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:

split_in_aaa
ms_in_aa
U5_aaa(x0)
U1_aa(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.

(51) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MS_IN_AA → U1_AA(split_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

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

(52) Obligation:

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

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

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
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:

split_in_aaa
ms_in_aa
U5_aaa(x0)
U1_aa(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.

(53) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U1_AA(split_out_aaa) → U2_AA(ms_in_aa) at position [0] we obtained the following new rules [LPAR04]:

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

(54) Obligation:

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

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

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
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:

split_in_aaa
ms_in_aa
U5_aaa(x0)
U1_aa(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.

(55) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = MS_IN_AA evaluates to t =MS_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

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

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

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.

(56) FALSE

(57) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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

(58) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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)

(59) DependencyPairsProof (EQUIVALENT transformation)

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

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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)
MS_IN_AG(x1, x2)  =  MS_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_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
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
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
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)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(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)

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

(60) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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)
MS_IN_AG(x1, x2)  =  MS_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
SPLIT_IN_AAA(x1, x2, x3)  =  SPLIT_IN_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
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
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
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)
U6_AAG(x1, x2, x3, x4, x5, x6)  =  U6_AAG(x1, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(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)

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

(61) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 23 less nodes.

(62) Complex Obligation (AND)

(63) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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

(64) UsableRulesProof (EQUIVALENT transformation)

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

(65) Obligation:

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

(66) PiDPToQDPProof (SOUND transformation)

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

(67) Obligation:

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.

(68) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS_IN_GA(s(X)) → LESS_IN_GA(X)
  The graph contains the following edges 1 > 1

(69) TRUE

(70) Obligation:

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

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)))
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)) → 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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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_AAG(x1, x2, x3)  =  MERGE_IN_AAG(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)

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

(71) UsableRulesProof (EQUIVALENT transformation)

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

(72) Obligation:

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

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

The TRS R consists of the following rules:

less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
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)
MERGE_IN_AAG(x1, x2, x3)  =  MERGE_IN_AAG(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)

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

(73) PiDPToQDPProof (SOUND transformation)

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

(74) Obligation:

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

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

(75) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• MERGE_IN_AAG(.(X, Zs)) → U6_AAG(X, Zs, less_in_ga(X))
  The graph contains the following edges 1 > 1, 1 > 2

• MERGE_IN_AAG(.(Y, Zs)) → U8_AAG(Y, Zs, less_in_ga(Y))
  The graph contains the following edges 1 > 1, 1 > 2

• U6_AAG(X, Zs, less_out_ga) → MERGE_IN_AAG(Zs)
  The graph contains the following edges 2 >= 1

• U8_AAG(Y, Zs, less_out_ga) → MERGE_IN_AAG(Zs)
  The graph contains the following edges 2 >= 1

(76) TRUE

(77) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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

(78) UsableRulesProof (EQUIVALENT transformation)

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

(79) Obligation:

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

(80) PiDPToQDPProof (SOUND transformation)

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

(81) Obligation:

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.

(82) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

(83) FALSE

(84) Obligation:

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)
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))
U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) → MERGE_IN_AAA(.(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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)

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

(85) UsableRulesProof (EQUIVALENT transformation)

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

(86) Obligation:

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

The TRS R consists of the following rules:

less_in_aa(0, s(X1)) → less_out_aa(0, s(X1))
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
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
U8_AAA(x1, x2, x3, x4, x5, x6)  =  U8_AAA(x6)

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

(87) PiDPToQDPProof (SOUND transformation)

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

(88) Obligation:

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

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

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.

(89) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE_IN_AAA → U6_AAA(less_in_aa) at position [0] we obtained the following new rules [LPAR04]:

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

(90) Obligation:

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

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

(91) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE_IN_AAA → U8_AAA(less_in_aa) at position [0] we obtained the following new rules [LPAR04]:

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

(92) Obligation:

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

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

(93) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = MERGE_IN_AAA evaluates to t =MERGE_IN_AAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

MERGE_IN_AAA → U6_AAA(less_out_aa(0))
with rule MERGE_IN_AAA → U6_AAA(less_out_aa(0)) at position [] and matcher [ ]

U6_AAA(less_out_aa(0)) → MERGE_IN_AAA
with rule U6_AAA(less_out_aa(X)) → MERGE_IN_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.

(94) FALSE

(95) Obligation:

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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

(96) UsableRulesProof (EQUIVALENT transformation)

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

(97) Obligation:

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

(98) PiDPToQDPProof (SOUND transformation)

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

(99) Obligation:

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.

(100) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

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

(101) FALSE

(102) Obligation:

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

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

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(X1)) → less_out_aa(0, s(X1))
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(X1)) → less_out_ga(0, s(X1))
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)
MS_IN_AA(x1, x2)  =  MS_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)

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

(103) UsableRulesProof (EQUIVALENT transformation)

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

(104) Obligation:

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

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

The TRS R consists of the following rules:

split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
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))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
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))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
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))
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(X1)) → less_out_aa(0, s(X1))
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)
MS_IN_AA(x1, x2)  =  MS_IN_AA
U1_AA(x1, x2, x3, x4, x5)  =  U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6)  =  U2_AA(x6)

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

(105) PiDPToQDPProof (SOUND transformation)

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

(106) Obligation:

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

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

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
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:

split_in_aaa
ms_in_aa
U5_aaa(x0)
U1_aa(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.

(107) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MS_IN_AA → U1_AA(split_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

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

(108) Obligation:

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

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

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
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:

split_in_aaa
ms_in_aa
U5_aaa(x0)
U1_aa(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.

(109) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U1_AA(split_out_aaa) → U2_AA(ms_in_aa) at position [0] we obtained the following new rules [LPAR04]:

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

(110) Obligation:

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

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

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
ms_in_aa → ms_out_aa
ms_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(ms_in_aa)
split_in_aaa → split_out_aaa
U2_aa(ms_out_aa) → U3_aa(ms_in_aa)
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:

split_in_aaa
ms_in_aa
U5_aaa(x0)
U1_aa(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.

(111) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = MS_IN_AA evaluates to t =MS_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

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

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.

(112) FALSE