(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).
Queries:
mergesort(g,a).
(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:
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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
x6)
MERGESORT_IN_GA(
x1,
x2) =
MERGESORT_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x1,
x2,
x3,
x5)
SPLIT_IN_GAA(
x1,
x2,
x3) =
SPLIT_IN_GAA(
x1)
U9_GAA(
x1,
x2,
x3,
x4,
x5) =
U9_GAA(
x1,
x2,
x5)
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GA(
x1,
x2,
x3,
x5,
x6)
U3_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_GA(
x1,
x2,
x3,
x5,
x6)
U4_GA(
x1,
x2,
x3,
x4,
x5) =
U4_GA(
x1,
x2,
x3,
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(
x1,
x2,
x3)
U6_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_GGA(
x1,
x2,
x3,
x4,
x6)
U7_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_GGA(
x1,
x2,
x3,
x4,
x6)
GT_IN_GG(
x1,
x2) =
GT_IN_GG(
x1,
x2)
U10_GG(
x1,
x2,
x3) =
U10_GG(
x1,
x2,
x3)
U8_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_GGA(
x1,
x2,
x3,
x4,
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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
x6)
MERGESORT_IN_GA(
x1,
x2) =
MERGESORT_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x1,
x2,
x3,
x5)
SPLIT_IN_GAA(
x1,
x2,
x3) =
SPLIT_IN_GAA(
x1)
U9_GAA(
x1,
x2,
x3,
x4,
x5) =
U9_GAA(
x1,
x2,
x5)
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GA(
x1,
x2,
x3,
x5,
x6)
U3_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_GA(
x1,
x2,
x3,
x5,
x6)
U4_GA(
x1,
x2,
x3,
x4,
x5) =
U4_GA(
x1,
x2,
x3,
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(
x1,
x2,
x3)
U6_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_GGA(
x1,
x2,
x3,
x4,
x6)
U7_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_GGA(
x1,
x2,
x3,
x4,
x6)
GT_IN_GG(
x1,
x2) =
GT_IN_GG(
x1,
x2)
U10_GG(
x1,
x2,
x3) =
U10_GG(
x1,
x2,
x3)
U8_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_GGA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
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) TRUE
(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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
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) TRUE
(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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
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(A, B)) → 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(A, B)) → MERGE_IN_GGA(.(A, X), Y)
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 set Q consists of the following terms:
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(26) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
U7_GGA(A, X, B, Y, gt_out_gg(A, B)) → MERGE_IN_GGA(.(A, X), Y)
Strictly oriented rules of the TRS R:
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
Used ordering: Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 2·x1 + x2
POL(0) = 2
POL(MERGE_IN_GGA(x1, x2)) = 2·x1 + 2·x2
POL(U10_gg(x1, x2, x3)) = 2·x1 + 2·x2 + x3
POL(U11_gg(x1, x2, x3)) = x1 + x2 + x3
POL(U5_GGA(x1, x2, x3, x4, x5)) = x1 + 2·x2 + 2·x3 + 2·x4 + 2·x5
POL(U7_GGA(x1, x2, x3, x4, x5)) = 2·x1 + 2·x2 + 2·x3 + 2·x4 + x5
POL(gt_in_gg(x1, x2)) = 2·x1 + 2·x2
POL(gt_out_gg(x1, x2)) = 1 + 2·x1 + x2
POL(le_in_gg(x1, x2)) = x1 + x2
POL(le_out_gg(x1, x2)) = x1 + x2
POL(s(x1)) = 2·x1
(27) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U5_GGA(A, X, B, Y, le_out_gg(A, B)) → 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(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))
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 set Q consists of the following terms:
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
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(A, B)) → MERGE_IN_GGA(X, .(B, Y))
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))
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 set Q consists of the following terms:
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
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(A, B)) → MERGE_IN_GGA(X, .(B, Y))
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)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
The set Q consists of the following terms:
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
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, x1, x2)
(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(A, B)) → MERGE_IN_GGA(X, .(B, Y))
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)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
The set Q consists of the following terms:
le_in_gg(x0, x1)
U11_gg(x0, x1, x2)
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(A, B)) → 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) TRUE
(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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
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) TRUE
(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(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x1,
x2,
x3,
x5)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U9_gaa(
x1,
x2,
x3,
x4,
x5) =
U9_gaa(
x1,
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x5,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U3_ga(
x1,
x2,
x3,
x5,
x6)
U4_ga(
x1,
x2,
x3,
x4,
x5) =
U4_ga(
x1,
x2,
x3,
x5)
merge_in_gga(
x1,
x2,
x3) =
merge_in_gga(
x1,
x2)
merge_out_gga(
x1,
x2,
x3) =
merge_out_gga(
x1,
x2,
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(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U6_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_gga(
x1,
x2,
x3,
x4,
x6)
U7_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U7_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U10_gg(
x1,
x2,
x3) =
U10_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gga(
x1,
x2,
x3,
x4,
x6)
MERGESORT_IN_GA(
x1,
x2) =
MERGESORT_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x1,
x2,
x3,
x5)
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GA(
x1,
x2,
x3,
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(E, F, U, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_GA(E, F, U, Y, mergesort_in_ga(W))
U2_GA(E, F, U, Y, mergesort_out_ga(W, X)) → MERGESORT_IN_GA(Y)
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(E, F, U, split_in_gaa(.(E, .(F, U))))
U1_GA(E, F, U, split_out_gaa(.(E, .(F, U)), 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, []), .(E, []))
mergesort_in_ga(.(E, .(F, U))) → U1_ga(E, F, U, split_in_gaa(.(E, .(F, U))))
split_in_gaa([]) → split_out_gaa([], [], [])
split_in_gaa(.(E, U)) → U9_gaa(E, U, split_in_gaa(U))
U9_gaa(E, U, split_out_gaa(U, W, V)) → split_out_gaa(.(E, U), .(E, V), W)
U1_ga(E, F, U, split_out_gaa(.(E, .(F, U)), W, Y)) → U2_ga(E, F, U, Y, mergesort_in_ga(W))
U2_ga(E, F, U, Y, mergesort_out_ga(W, X)) → U3_ga(E, F, U, X, mergesort_in_ga(Y))
U3_ga(E, F, U, X, mergesort_out_ga(Y, Z)) → U4_ga(E, F, U, merge_in_gga(X, Z))
merge_in_gga(X, []) → merge_out_gga(X, [], X)
merge_in_gga([], X) → merge_out_gga([], X, 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(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, le_out_gg(A, B)) → U6_gga(A, X, B, Y, 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(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, gt_out_gg(A, B)) → U8_gga(A, X, B, Y, merge_in_gga(.(A, X), Y))
U8_gga(A, X, B, Y, merge_out_gga(.(A, X), Y, Z)) → merge_out_gga(.(A, X), .(B, Y), .(B, Z))
U6_gga(A, X, B, Y, merge_out_gga(X, .(B, Y), Z)) → merge_out_gga(.(A, X), .(B, Y), .(A, Z))
U4_ga(E, F, U, merge_out_gga(X, Z, V)) → mergesort_out_ga(.(E, .(F, U)), V)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U9_gaa(x0, x1, x2)
U1_ga(x0, x1, x2, x3)
U2_ga(x0, x1, x2, x3, x4)
U3_ga(x0, x1, x2, x3, x4)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0, x1, x2)
U5_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0, x1, x2)
U7_gga(x0, x1, x2, x3, x4)
U8_gga(x0, x1, x2, x3, x4)
U6_gga(x0, x1, x2, x3, x4)
U4_ga(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(46) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. 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
(47) 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)
(48) 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
(49) 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
(50) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 11 less nodes.
(51) Complex Obligation (AND)
(52) 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
(53) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(54) 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
(55) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(56) 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.
(57) 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
(58) TRUE
(59) 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
(60) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(61) 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
(62) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(63) 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.
(64) 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
(65) TRUE
(66) 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
(67) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(68) 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
(69) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(70) 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.
(71) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U5_GGA(A, X, B, Y, le_out_gg) → MERGE_IN_GGA(X, .(B, Y))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(.(x1, x2)) = x1 + x2
POL(0) = 1
POL(MERGE_IN_GGA(x1, x2)) = x1
POL(U10_gg(x1)) = 0
POL(U11_gg(x1)) = 1
POL(U5_GGA(x1, x2, x3, x4, x5)) = x2 + x5
POL(U7_GGA(x1, x2, x3, x4, x5)) = x1 + x2
POL(gt_in_gg(x1, x2)) = 0
POL(gt_out_gg) = 0
POL(le_in_gg(x1, x2)) = x1
POL(le_out_gg) = 1
POL(s(x1)) = 1
The following usable rules [FROCOS05] were oriented:
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
(72) 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))
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.
(73) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(74) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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.
(75) 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.
(76) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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:
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
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.
(77) 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].
le_in_gg(x0, x1)
U11_gg(x0)
(78) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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:
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
The set Q consists of the following terms:
gt_in_gg(x0, x1)
U10_gg(x0)
We have to consider all (P,Q,R)-chains.
(79) 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:
- U7_GGA(A, X, B, Y, gt_out_gg) → MERGE_IN_GGA(.(A, X), Y)
The graph contains the following edges 4 >= 2
- MERGE_IN_GGA(.(A, X), .(B, Y)) → U7_GGA(A, X, B, Y, gt_in_gg(A, B))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4
(80) TRUE
(81) 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
(82) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(83) 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
(84) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(85) 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.
(86) 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
(87) TRUE
(88) 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
(89) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(90) 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.
(91) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
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: Matrix interpretation [MATRO]:
| POL(split_out_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U2_GA(x1, x2)) = | 0 | + | | · | x1 | + | | · | x2 |
| POL(mergesort_in_ga(x1)) = | | + | | · | x1 |
| POL(mergesort_out_ga(x1)) = | | + | | · | x1 |
| POL(MERGESORT_IN_GA(x1)) = | 0 | + | | · | x1 |
| POL(.(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split_in_gaa(x1)) = | | + | | · | x1 |
| POL(U9_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U2_ga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U3_ga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(merge_in_gga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(merge_out_gga(x1)) = | | + | | · | x1 |
| POL(U5_gga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(le_in_gg(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U7_gga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(gt_in_gg(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U6_gga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U8_gga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
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(.(E, U)) → U9_gaa(E, split_in_gaa(U))
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))
split_in_gaa([]) → split_out_gaa([], [])
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
U4_ga(merge_out_gga(V)) → mergesort_out_ga(V)
(92) 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))
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.
(93) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(94) 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(split_in_gaa(.(E, .(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.
(95) 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.
(96) 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(split_in_gaa(.(E, .(F, U))))
The TRS R consists of the following rules:
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
split_in_gaa([]) → split_out_gaa([], [])
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.
(97) 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)
(98) 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(split_in_gaa(.(E, .(F, U))))
The TRS R consists of the following rules:
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
split_in_gaa([]) → split_out_gaa([], [])
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.
(99) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U1_GA(split_out_gaa(W, Y)) → MERGESORT_IN_GA(W)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
| POL(split_out_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(MERGESORT_IN_GA(x1)) = | 0 | + | | · | x1 |
| POL(.(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split_in_gaa(x1)) = | | + | | · | x1 |
| POL(U9_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
split_in_gaa([]) → split_out_gaa([], [])
U9_gaa(E, split_out_gaa(W, V)) → split_out_gaa(.(E, V), W)
(100) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_GA(.(E, .(F, U))) → U1_GA(split_in_gaa(.(E, .(F, U))))
The TRS R consists of the following rules:
split_in_gaa(.(E, U)) → U9_gaa(E, split_in_gaa(U))
split_in_gaa([]) → split_out_gaa([], [])
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.
(101) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(102) TRUE