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

0 Prolog
↳1 PrologToPiTRSProof (⇒, 72 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 188 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇔, 18 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇔, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇒, 0 ms)
        ↳25 QDP
        ↳26 QDPOrderProof (⇔, 50 ms)
        ↳27 QDP
        ↳28 DependencyGraphProof (⇔, 0 ms)
        ↳29 QDP
        ↳30 UsableRulesProof (⇔, 0 ms)
        ↳31 QDP
        ↳32 QReductionProof (⇔, 0 ms)
        ↳33 QDP
        ↳34 QDPSizeChangeProof (⇔, 0 ms)
        ↳35 YES
    ↳36 PiDP
        ↳37 UsableRulesProof (⇔, 0 ms)
        ↳38 PiDP
        ↳39 PiDPToQDPProof (⇒, 0 ms)
        ↳40 QDP
        ↳41 QDPSizeChangeProof (⇔, 0 ms)
        ↳42 YES
    ↳43 PiDP
        ↳44 PiDPToQDPProof (⇒, 0 ms)
        ↳45 QDP
        ↳46 Rewriting (⇔, 28 ms)
        ↳47 QDP
        ↳48 Rewriting (⇔, 0 ms)
        ↳49 QDP
        ↳50 QDPQMonotonicMRRProof (⇔, 762 ms)
        ↳51 QDP
        ↳52 QDPOrderProof (⇔, 97 ms)
        ↳53 QDP
        ↳54 DependencyGraphProof (⇔, 0 ms)
        ↳55 QDP
        ↳56 UsableRulesProof (⇔, 0 ms)
        ↳57 QDP
        ↳58 QReductionProof (⇔, 0 ms)
        ↳59 QDP
        ↳60 QDPOrderProof (⇔, 156 ms)
        ↳61 QDP
        ↳62 DependencyGraphProof (⇔, 0 ms)
        ↳63 TRUE

(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(E, []), .(E, [])).
mergesort(.(E, .(F, U)), V) :- ','(split(.(E, .(F, U)), W, Y), ','(mergesort(W, X), ','(mergesort(Y, Z), merge(X, Z, V)))).
merge(X, [], X).
merge([], X, X).
merge(.(A, X), .(B, Y), .(A, Z)) :- ','(le(A, B), merge(X, .(B, Y), Z)).
merge(.(A, X), .(B, Y), .(B, Z)) :- ','(gt(A, B), merge(.(A, X), Y, Z)).
split([], [], []).
split(.(E, U), .(E, V), W) :- split(U, W, V).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Query: mergesort(g,a)

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort_in: (b,f)
split_in: (b,f,f)
merge_in: (b,b,f)
le_in: (b,b)
gt_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

MERGESORT_IN_GA(.(E, .(F, U)), V) → U1_GA(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
MERGESORT_IN_GA(.(E, .(F, U)), V) → SPLIT_IN_GAA(.(E, .(F, U)), W, Y)
SPLIT_IN_GAA(.(E, U), .(E, V), W) → U9_GAA(E, U, V, W, split_in_gaa(U, W, V))
SPLIT_IN_GAA(.(E, U), .(E, V), W) → SPLIT_IN_GAA(U, W, V)
U1_GA(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_GA(E, F, U, V, Y, mergesort_in_ga(W, X))
U1_GA(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → MERGESORT_IN_GA(W, X)
U2_GA(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_GA(E, F, U, V, X, mergesort_in_ga(Y, Z))
U2_GA(E, F, U, V, Y, mergesort_out_ga(W, X)) → MERGESORT_IN_GA(Y, Z)
U3_GA(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_GA(E, F, U, V, merge_in_gga(X, Z, V))
U3_GA(E, F, U, V, X, mergesort_out_ga(Y, Z)) → MERGE_IN_GGA(X, Z, V)
MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) → U5_GGA(A, X, B, Y, Z, le_in_gg(A, B))
MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) → LE_IN_GG(A, B)
LE_IN_GG(s(X), s(Y)) → U11_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U5_GGA(A, X, B, Y, Z, le_out_gg(A, B)) → U6_GGA(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
U5_GGA(A, X, B, Y, Z, le_out_gg(A, B)) → MERGE_IN_GGA(X, .(B, Y), Z)
MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) → U7_GGA(A, X, B, Y, Z, gt_in_gg(A, B))
MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) → GT_IN_GG(A, B)
GT_IN_GG(s(X), s(Y)) → U10_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U7_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_GGA(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U7_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) → MERGE_IN_GGA(.(A, X), Y, Z)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)
MERGESORT_IN_GA(x1, x2)  =  MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x5, x6)
U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x5, x6)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U11_GG(x1, x2, x3)  =  U11_GG(x3)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x6)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x1, x2, x3, x4, x6)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
U10_GG(x1, x2, x3)  =  U10_GG(x3)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x3, x6)

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

(4) Obligation:

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

MERGESORT_IN_GA(.(E, .(F, U)), V) → U1_GA(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
MERGESORT_IN_GA(.(E, .(F, U)), V) → SPLIT_IN_GAA(.(E, .(F, U)), W, Y)
SPLIT_IN_GAA(.(E, U), .(E, V), W) → U9_GAA(E, U, V, W, split_in_gaa(U, W, V))
SPLIT_IN_GAA(.(E, U), .(E, V), W) → SPLIT_IN_GAA(U, W, V)
U1_GA(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_GA(E, F, U, V, Y, mergesort_in_ga(W, X))
U1_GA(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → MERGESORT_IN_GA(W, X)
U2_GA(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_GA(E, F, U, V, X, mergesort_in_ga(Y, Z))
U2_GA(E, F, U, V, Y, mergesort_out_ga(W, X)) → MERGESORT_IN_GA(Y, Z)
U3_GA(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_GA(E, F, U, V, merge_in_gga(X, Z, V))
U3_GA(E, F, U, V, X, mergesort_out_ga(Y, Z)) → MERGE_IN_GGA(X, Z, V)
MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) → U5_GGA(A, X, B, Y, Z, le_in_gg(A, B))
MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) → LE_IN_GG(A, B)
LE_IN_GG(s(X), s(Y)) → U11_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U5_GGA(A, X, B, Y, Z, le_out_gg(A, B)) → U6_GGA(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
U5_GGA(A, X, B, Y, Z, le_out_gg(A, B)) → MERGE_IN_GGA(X, .(B, Y), Z)
MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) → U7_GGA(A, X, B, Y, Z, gt_in_gg(A, B))
MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) → GT_IN_GG(A, B)
GT_IN_GG(s(X), s(Y)) → U10_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U7_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_GGA(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U7_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) → MERGE_IN_GGA(.(A, X), Y, Z)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)
MERGESORT_IN_GA(x1, x2)  =  MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x5, x6)
U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x5, x6)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U11_GG(x1, x2, x3)  =  U11_GG(x3)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x6)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x1, x2, x3, x4, x6)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
U10_GG(x1, x2, x3)  =  U10_GG(x3)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x3, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 11 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

U5_GGA(A, X, B, Y, Z, le_out_gg(A, B)) → MERGE_IN_GGA(X, .(B, Y), Z)
MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) → U5_GGA(A, X, B, Y, Z, le_in_gg(A, B))
MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) → U7_GGA(A, X, B, Y, Z, gt_in_gg(A, B))
U7_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) → MERGE_IN_GGA(.(A, X), Y, Z)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x1, x2, x3, x4, x6)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U5_GGA(A, X, B, Y, Z, le_out_gg(A, B)) → MERGE_IN_GGA(X, .(B, Y), Z)
MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) → U5_GGA(A, X, B, Y, Z, le_in_gg(A, B))
MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) → U7_GGA(A, X, B, Y, Z, gt_in_gg(A, B))
U7_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) → MERGE_IN_GGA(.(A, X), Y, Z)

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
MERGE_IN_GGA(x1, x2, x3)  =  MERGE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6)  =  U7_GGA(x1, x2, x3, x4, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

U5_GGA(A, X, B, Y, le_out_gg) → MERGE_IN_GGA(X, .(B, Y))
MERGE_IN_GGA(.(A, X), .(B, Y)) → U5_GGA(A, X, B, Y, le_in_gg(A, B))
MERGE_IN_GGA(.(A, X), .(B, Y)) → U7_GGA(A, X, B, Y, gt_in_gg(A, B))
U7_GGA(A, X, B, Y, gt_out_gg) → MERGE_IN_GGA(.(A, X), Y)

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
U10_gg(gt_out_gg) → gt_out_gg

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


U7_GGA(A, X, B, Y, gt_out_gg) → MERGE_IN_GGA(.(A, X), Y)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(0) = 0   
POL(MERGE_IN_GGA(x1, x2)) = x2   
POL(U10_gg(x1)) = x1   
POL(U11_gg(x1)) = x1   
POL(U5_GGA(x1, x2, x3, x4, x5)) = 1 + x4   
POL(U7_GGA(x1, x2, x3, x4, x5)) = 1 + x4   
POL(gt_in_gg(x1, x2)) = x2   
POL(gt_out_gg) = 0   
POL(le_in_gg(x1, x2)) = x1   
POL(le_out_gg) = 0   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(27) Obligation:

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

U5_GGA(A, X, B, Y, le_out_gg) → MERGE_IN_GGA(X, .(B, Y))
MERGE_IN_GGA(.(A, X), .(B, Y)) → U5_GGA(A, X, B, Y, le_in_gg(A, B))
MERGE_IN_GGA(.(A, X), .(B, Y)) → U7_GGA(A, X, B, Y, gt_in_gg(A, B))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
U10_gg(gt_out_gg) → gt_out_gg

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) Obligation:

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

MERGE_IN_GGA(.(A, X), .(B, Y)) → U5_GGA(A, X, B, Y, le_in_gg(A, B))
U5_GGA(A, X, B, Y, le_out_gg) → MERGE_IN_GGA(X, .(B, Y))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
U10_gg(gt_out_gg) → gt_out_gg

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)

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

(30) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(31) Obligation:

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

MERGE_IN_GGA(.(A, X), .(B, Y)) → U5_GGA(A, X, B, Y, le_in_gg(A, B))
U5_GGA(A, X, B, Y, le_out_gg) → MERGE_IN_GGA(X, .(B, Y))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)

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

(32) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

gt_in_gg(x0, x1)
U10_gg(x0)

(33) Obligation:

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

MERGE_IN_GGA(.(A, X), .(B, Y)) → U5_GGA(A, X, B, Y, le_in_gg(A, B))
U5_GGA(A, X, B, Y, le_out_gg) → MERGE_IN_GGA(X, .(B, Y))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg

The set Q consists of the following terms:

le_in_gg(x0, x1)
U11_gg(x0)

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

(34) 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:

  • U5_GGA(A, X, B, Y, le_out_gg) → MERGE_IN_GGA(X, .(B, Y))
    The graph contains the following edges 2 >= 1

  • MERGE_IN_GGA(.(A, X), .(B, Y)) → U5_GGA(A, X, B, Y, le_in_gg(A, B))
    The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4

(35) YES

(36) Obligation:

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

SPLIT_IN_GAA(.(E, U), .(E, V), W) → SPLIT_IN_GAA(U, W, V)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

SPLIT_IN_GAA(.(E, U), .(E, V), W) → SPLIT_IN_GAA(U, W, V)

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

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

SPLIT_IN_GAA(.(E, U)) → SPLIT_IN_GAA(U)

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

(41) 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:

  • SPLIT_IN_GAA(.(E, U)) → SPLIT_IN_GAA(U)
    The graph contains the following edges 1 > 1

(42) YES

(43) Obligation:

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

U1_GA(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_GA(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_GA(E, F, U, V, Y, mergesort_out_ga(W, X)) → MERGESORT_IN_GA(Y, Z)
MERGESORT_IN_GA(.(E, .(F, U)), V) → U1_GA(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
U1_GA(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → MERGESORT_IN_GA(W, X)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(E, []), .(E, [])) → mergesort_out_ga(.(E, []), .(E, []))
mergesort_in_ga(.(E, .(F, U)), V) → U1_ga(E, F, U, V, split_in_gaa(.(E, .(F, U)), W, Y))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(E, U), .(E, V), W) → U9_gaa(E, U, V, W, split_in_gaa(U, W, V))
U9_gaa(E, U, V, W, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, V, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, V, Y, mergesort_in_ga(W, X))
U2_ga(E, F, U, V, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, V, X, mergesort_in_ga(Y, Z))
U3_ga(E, F, U, V, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, V, merge_in_gga(X, Z, V))
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(.(A, X), .(B, Y), .(A, Z)) → U5_gga(A, X, B, Y, Z, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U5_gga(A, X, B, Y, Z, le_out_gg(A, B)) → U6_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z))
merge_in_gga(.(A, X), .(B, Y), .(B, Z)) → U7_gga(A, X, B, Y, Z, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U7_gga(A, X, B, Y, Z, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z))
U8_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, V, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2)  =  mergesort_in_ga(x1)
[]  =  []
mergesort_out_ga(x1, x2)  =  mergesort_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
merge_in_gga(x1, x2, x3)  =  merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3)  =  merge_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x6)
U7_gga(x1, x2, x3, x4, x5, x6)  =  U7_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U8_gga(x1, x2, x3, x4, x5, x6)  =  U8_gga(x3, x6)
MERGESORT_IN_GA(x1, x2)  =  MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x5, x6)

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

(44) PiDPToQDPProof (SOUND transformation)

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

(45) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → U2_GA(Y, mergesort_in_ga(W))
U2_GA(Y, mergesort_out_ga(X)) → MERGESORT_IN_GA(Y)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(split_in_gaa(.(E, .(F, U))))
U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(E, [])) → mergesort_out_ga(.(E, []))
mergesort_in_ga(.(E, .(F, U))) → U1_ga(split_in_gaa(.(E, .(F, U))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
U1_ga(split_out_gaa(W, Y)) → U2_ga(Y, mergesort_in_ga(W))
U2_ga(Y, mergesort_out_ga(X)) → U3_ga(X, mergesort_in_ga(Y))
U3_ga(X, mergesort_out_ga(Z)) → U4_ga(merge_in_gga(X, Z))
merge_in_gga(X, []) → merge_out_gga(X)
merge_in_gga([], X) → merge_out_gga(X)
merge_in_gga(.(A, X), .(B, Y)) → U5_gga(A, X, B, Y, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U5_gga(A, X, B, Y, le_out_gg) → U6_gga(A, merge_in_gga(X, .(B, Y)))
merge_in_gga(.(A, X), .(B, Y)) → U7_gga(A, X, B, Y, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U7_gga(A, X, B, Y, gt_out_gg) → U8_gga(B, merge_in_gga(.(A, X), Y))
U8_gga(B, merge_out_gga(Z)) → merge_out_gga(.(B, Z))
U6_gga(A, merge_out_gga(Z)) → merge_out_gga(.(A, Z))
U4_ga(merge_out_gga(V)) → mergesort_out_ga(V)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U9_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1)
U6_gga(x0, x1)
U4_ga(x0)

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

(46) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(split_in_gaa(.(E, .(F, U)))) at position [0] we obtained the following new rules [LPAR04]:

MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, split_in_gaa(.(F, U))))

(47) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → U2_GA(Y, mergesort_in_ga(W))
U2_GA(Y, mergesort_out_ga(X)) → MERGESORT_IN_GA(Y)
U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, split_in_gaa(.(F, U))))

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(E, [])) → mergesort_out_ga(.(E, []))
mergesort_in_ga(.(E, .(F, U))) → U1_ga(split_in_gaa(.(E, .(F, U))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
U1_ga(split_out_gaa(W, Y)) → U2_ga(Y, mergesort_in_ga(W))
U2_ga(Y, mergesort_out_ga(X)) → U3_ga(X, mergesort_in_ga(Y))
U3_ga(X, mergesort_out_ga(Z)) → U4_ga(merge_in_gga(X, Z))
merge_in_gga(X, []) → merge_out_gga(X)
merge_in_gga([], X) → merge_out_gga(X)
merge_in_gga(.(A, X), .(B, Y)) → U5_gga(A, X, B, Y, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U5_gga(A, X, B, Y, le_out_gg) → U6_gga(A, merge_in_gga(X, .(B, Y)))
merge_in_gga(.(A, X), .(B, Y)) → U7_gga(A, X, B, Y, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U7_gga(A, X, B, Y, gt_out_gg) → U8_gga(B, merge_in_gga(.(A, X), Y))
U8_gga(B, merge_out_gga(Z)) → merge_out_gga(.(B, Z))
U6_gga(A, merge_out_gga(Z)) → merge_out_gga(.(A, Z))
U4_ga(merge_out_gga(V)) → mergesort_out_ga(V)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U9_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1)
U6_gga(x0, x1)
U4_ga(x0)

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

(48) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, split_in_gaa(.(F, U)))) at position [0,1] we obtained the following new rules [LPAR04]:

MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, U9_gaa(F, split_in_gaa(U))))

(49) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → U2_GA(Y, mergesort_in_ga(W))
U2_GA(Y, mergesort_out_ga(X)) → MERGESORT_IN_GA(Y)
U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, U9_gaa(F, split_in_gaa(U))))

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(E, [])) → mergesort_out_ga(.(E, []))
mergesort_in_ga(.(E, .(F, U))) → U1_ga(split_in_gaa(.(E, .(F, U))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
U1_ga(split_out_gaa(W, Y)) → U2_ga(Y, mergesort_in_ga(W))
U2_ga(Y, mergesort_out_ga(X)) → U3_ga(X, mergesort_in_ga(Y))
U3_ga(X, mergesort_out_ga(Z)) → U4_ga(merge_in_gga(X, Z))
merge_in_gga(X, []) → merge_out_gga(X)
merge_in_gga([], X) → merge_out_gga(X)
merge_in_gga(.(A, X), .(B, Y)) → U5_gga(A, X, B, Y, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U5_gga(A, X, B, Y, le_out_gg) → U6_gga(A, merge_in_gga(X, .(B, Y)))
merge_in_gga(.(A, X), .(B, Y)) → U7_gga(A, X, B, Y, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U7_gga(A, X, B, Y, gt_out_gg) → U8_gga(B, merge_in_gga(.(A, X), Y))
U8_gga(B, merge_out_gga(Z)) → merge_out_gga(.(B, Z))
U6_gga(A, merge_out_gga(Z)) → merge_out_gga(.(A, Z))
U4_ga(merge_out_gga(V)) → mergesort_out_ga(V)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U9_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1)
U6_gga(x0, x1)
U4_ga(x0)

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

(50) QDPQMonotonicMRRProof (EQUIVALENT transformation)

By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented rules of the TRS R:

mergesort_in_ga(.(E, [])) → mergesort_out_ga(.(E, []))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 2 + 2·x2   
POL(0) = 0   
POL(MERGESORT_IN_GA(x1)) = 2·x1   
POL(U10_gg(x1)) = 0   
POL(U11_gg(x1)) = 1   
POL(U1_GA(x1)) = 2·x1   
POL(U1_ga(x1)) = 2·x1   
POL(U2_GA(x1, x2)) = 2·x1 + x2   
POL(U2_ga(x1, x2)) = 2·x1 + x2   
POL(U3_ga(x1, x2)) = x2   
POL(U4_ga(x1)) = 0   
POL(U5_gga(x1, x2, x3, x4, x5)) = 2·x4   
POL(U6_gga(x1, x2)) = 0   
POL(U7_gga(x1, x2, x3, x4, x5)) = 2·x4   
POL(U8_gga(x1, x2)) = 0   
POL(U9_gaa(x1, x2)) = 2 + 2·x2   
POL([]) = 0   
POL(gt_in_gg(x1, x2)) = 0   
POL(gt_out_gg) = 0   
POL(le_in_gg(x1, x2)) = 2   
POL(le_out_gg) = 0   
POL(merge_in_gga(x1, x2)) = 2·x2   
POL(merge_out_gga(x1)) = 0   
POL(mergesort_in_ga(x1)) = 2·x1   
POL(mergesort_out_ga(x1)) = 0   
POL(s(x1)) = 0   
POL(split_in_gaa(x1)) = x1   
POL(split_out_gaa(x1, x2)) = x1 + x2   

(51) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → U2_GA(Y, mergesort_in_ga(W))
U2_GA(Y, mergesort_out_ga(X)) → MERGESORT_IN_GA(Y)
U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, U9_gaa(F, split_in_gaa(U))))

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(E, .(F, U))) → U1_ga(split_in_gaa(.(E, .(F, U))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
U1_ga(split_out_gaa(W, Y)) → U2_ga(Y, mergesort_in_ga(W))
U2_ga(Y, mergesort_out_ga(X)) → U3_ga(X, mergesort_in_ga(Y))
U3_ga(X, mergesort_out_ga(Z)) → U4_ga(merge_in_gga(X, Z))
merge_in_gga(X, []) → merge_out_gga(X)
merge_in_gga([], X) → merge_out_gga(X)
merge_in_gga(.(A, X), .(B, Y)) → U5_gga(A, X, B, Y, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U5_gga(A, X, B, Y, le_out_gg) → U6_gga(A, merge_in_gga(X, .(B, Y)))
merge_in_gga(.(A, X), .(B, Y)) → U7_gga(A, X, B, Y, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U7_gga(A, X, B, Y, gt_out_gg) → U8_gga(B, merge_in_gga(.(A, X), Y))
U8_gga(B, merge_out_gga(Z)) → merge_out_gga(.(B, Z))
U6_gga(A, merge_out_gga(Z)) → merge_out_gga(.(A, Z))
U4_ga(merge_out_gga(V)) → mergesort_out_ga(V)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U9_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1)
U6_gga(x0, x1)
U4_ga(x0)

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


U2_GA(Y, mergesort_out_ga(X)) → MERGESORT_IN_GA(Y)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 0   
POL(0) = 1   
POL(MERGESORT_IN_GA(x1)) = 0   
POL(U10_gg(x1)) = 0   
POL(U11_gg(x1)) = 1   
POL(U1_GA(x1)) = x1   
POL(U1_ga(x1)) = x1   
POL(U2_GA(x1, x2)) = x2   
POL(U2_ga(x1, x2)) = x2   
POL(U3_ga(x1, x2)) = 1   
POL(U4_ga(x1)) = 1   
POL(U5_gga(x1, x2, x3, x4, x5)) = 0   
POL(U6_gga(x1, x2)) = 0   
POL(U7_gga(x1, x2, x3, x4, x5)) = 0   
POL(U8_gga(x1, x2)) = 0   
POL(U9_gaa(x1, x2)) = 0   
POL([]) = 1   
POL(gt_in_gg(x1, x2)) = 0   
POL(gt_out_gg) = 0   
POL(le_in_gg(x1, x2)) = 1 + x1   
POL(le_out_gg) = 1   
POL(merge_in_gga(x1, x2)) = 0   
POL(merge_out_gga(x1)) = 0   
POL(mergesort_in_ga(x1)) = x1   
POL(mergesort_out_ga(x1)) = 1   
POL(s(x1)) = 1 + x1   
POL(split_in_gaa(x1)) = 0   
POL(split_out_gaa(x1, x2)) = x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(E, .(F, U))) → U1_ga(split_in_gaa(.(E, .(F, U))))
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
U1_ga(split_out_gaa(W, Y)) → U2_ga(Y, mergesort_in_ga(W))
U2_ga(Y, mergesort_out_ga(X)) → U3_ga(X, mergesort_in_ga(Y))
U3_ga(X, mergesort_out_ga(Z)) → U4_ga(merge_in_gga(X, Z))
U4_ga(merge_out_gga(V)) → mergesort_out_ga(V)

(53) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → U2_GA(Y, mergesort_in_ga(W))
U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, U9_gaa(F, split_in_gaa(U))))

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(E, .(F, U))) → U1_ga(split_in_gaa(.(E, .(F, U))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
U1_ga(split_out_gaa(W, Y)) → U2_ga(Y, mergesort_in_ga(W))
U2_ga(Y, mergesort_out_ga(X)) → U3_ga(X, mergesort_in_ga(Y))
U3_ga(X, mergesort_out_ga(Z)) → U4_ga(merge_in_gga(X, Z))
merge_in_gga(X, []) → merge_out_gga(X)
merge_in_gga([], X) → merge_out_gga(X)
merge_in_gga(.(A, X), .(B, Y)) → U5_gga(A, X, B, Y, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U5_gga(A, X, B, Y, le_out_gg) → U6_gga(A, merge_in_gga(X, .(B, Y)))
merge_in_gga(.(A, X), .(B, Y)) → U7_gga(A, X, B, Y, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U7_gga(A, X, B, Y, gt_out_gg) → U8_gga(B, merge_in_gga(.(A, X), Y))
U8_gga(B, merge_out_gga(Z)) → merge_out_gga(.(B, Z))
U6_gga(A, merge_out_gga(Z)) → merge_out_gga(.(A, Z))
U4_ga(merge_out_gga(V)) → mergesort_out_ga(V)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U9_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1)
U6_gga(x0, x1)
U4_ga(x0)

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

(54) DependencyGraphProof (EQUIVALENT transformation)

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

(55) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, U9_gaa(F, split_in_gaa(U))))

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(E, .(F, U))) → U1_ga(split_in_gaa(.(E, .(F, U))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
U1_ga(split_out_gaa(W, Y)) → U2_ga(Y, mergesort_in_ga(W))
U2_ga(Y, mergesort_out_ga(X)) → U3_ga(X, mergesort_in_ga(Y))
U3_ga(X, mergesort_out_ga(Z)) → U4_ga(merge_in_gga(X, Z))
merge_in_gga(X, []) → merge_out_gga(X)
merge_in_gga([], X) → merge_out_gga(X)
merge_in_gga(.(A, X), .(B, Y)) → U5_gga(A, X, B, Y, le_in_gg(A, B))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U5_gga(A, X, B, Y, le_out_gg) → U6_gga(A, merge_in_gga(X, .(B, Y)))
merge_in_gga(.(A, X), .(B, Y)) → U7_gga(A, X, B, Y, gt_in_gg(A, B))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U7_gga(A, X, B, Y, gt_out_gg) → U8_gga(B, merge_in_gga(.(A, X), Y))
U8_gga(B, merge_out_gga(Z)) → merge_out_gga(.(B, Z))
U6_gga(A, merge_out_gga(Z)) → merge_out_gga(.(A, Z))
U4_ga(merge_out_gga(V)) → mergesort_out_ga(V)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U9_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1)
U6_gga(x0, x1)
U4_ga(x0)

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

(56) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(57) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, U9_gaa(F, split_in_gaa(U))))

The TRS R consists of the following rules:

split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U9_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1)
U6_gga(x0, x1)
U4_ga(x0)

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

(58) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

mergesort_in_ga(x0)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1)
U6_gga(x0, x1)
U4_ga(x0)

(59) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, U9_gaa(F, split_in_gaa(U))))

The TRS R consists of the following rules:

split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)

The set Q consists of the following terms:

split_in_gaa(x0)
U9_gaa(x0, x1)

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(U9_gaa(E, U9_gaa(F, split_in_gaa(U))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U1_GA(x1)) = -I + 0A·x1

POL(split_out_gaa(x1, x2)) = 0A + 0A·x1 + 3A·x2

POL(MERGESORT_IN_GA(x1)) = 0A + 0A·x1

POL(.(x1, x2)) = -I + -I·x1 + 5A·x2

POL(U9_gaa(x1, x2)) = -I + -I·x1 + 3A·x2

POL(split_in_gaa(x1)) = -I + 3A·x1

POL([]) = 0A

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)

(61) Obligation:

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

U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)

The TRS R consists of the following rules:

split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)

The set Q consists of the following terms:

split_in_gaa(x0)
U9_gaa(x0, x1)

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

(62) DependencyGraphProof (EQUIVALENT transformation)

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

(63) TRUE