(0) Obligation:
Clauses:
mergesort([], []).
mergesort(.(X, []), .(X, [])).
mergesort(.(X, .(Y, L1)), L2) :- ','(split2(.(X, .(Y, L1)), L3, L4), ','(mergesort(L3, L5), ','(mergesort(L4, L6), merge(L5, L6, L2)))).
split(L1, L2, L3) :- split0(L1, L2, L3).
split(L1, L2, L3) :- split1(L1, L2, L3).
split(L1, L2, L3) :- split2(L1, L2, L3).
split0([], [], []).
split1(.(X, []), .(X, []), []).
split2(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) :- split(L1, L2, L3).
merge([], L1, L1).
merge(L1, [], L1).
merge(.(X, L1), .(Y, L2), .(X, L3)) :- ','(le(X, Y), merge(L1, .(Y, L2), L3)).
merge(.(X, L1), .(Y, L2), .(Y, L3)) :- ','(gt(X, Y), merge(.(X, L1), L2, L3)).
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)
split2_in: (b,f,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(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(2) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4)
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT0_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT1_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → MERGE_IN_GGA(L5, L6, L2)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U14_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_GGA(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U13_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_GGA(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
MERGESORT_IN_GA(
x1,
x2) =
MERGESORT_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x5)
SPLIT2_IN_GAA(
x1,
x2,
x3) =
SPLIT2_IN_GAA(
x1)
U8_GAA(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_GAA(
x1,
x2,
x6)
SPLIT_IN_GAA(
x1,
x2,
x3) =
SPLIT_IN_GAA(
x1)
U5_GAA(
x1,
x2,
x3,
x4) =
U5_GAA(
x4)
SPLIT0_IN_GAA(
x1,
x2,
x3) =
SPLIT0_IN_GAA(
x1)
U6_GAA(
x1,
x2,
x3,
x4) =
U6_GAA(
x4)
SPLIT1_IN_GAA(
x1,
x2,
x3) =
SPLIT1_IN_GAA(
x1)
U7_GAA(
x1,
x2,
x3,
x4) =
U7_GAA(
x4)
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)
U9_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_GGA(
x1,
x2,
x3,
x4,
x6)
LE_IN_GG(
x1,
x2) =
LE_IN_GG(
x1,
x2)
U14_GG(
x1,
x2,
x3) =
U14_GG(
x3)
U10_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_GGA(
x1,
x6)
U11_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_GGA(
x1,
x2,
x3,
x4,
x6)
GT_IN_GG(
x1,
x2) =
GT_IN_GG(
x1,
x2)
U13_GG(
x1,
x2,
x3) =
U13_GG(
x3)
U12_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_GGA(
x3,
x6)
We have to consider all (P,R,Pi)-chains
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4)
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT0_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT1_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → MERGE_IN_GGA(L5, L6, L2)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U14_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_GGA(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U13_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_GGA(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
MERGESORT_IN_GA(
x1,
x2) =
MERGESORT_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x5)
SPLIT2_IN_GAA(
x1,
x2,
x3) =
SPLIT2_IN_GAA(
x1)
U8_GAA(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_GAA(
x1,
x2,
x6)
SPLIT_IN_GAA(
x1,
x2,
x3) =
SPLIT_IN_GAA(
x1)
U5_GAA(
x1,
x2,
x3,
x4) =
U5_GAA(
x4)
SPLIT0_IN_GAA(
x1,
x2,
x3) =
SPLIT0_IN_GAA(
x1)
U6_GAA(
x1,
x2,
x3,
x4) =
U6_GAA(
x4)
SPLIT1_IN_GAA(
x1,
x2,
x3) =
SPLIT1_IN_GAA(
x1)
U7_GAA(
x1,
x2,
x3,
x4) =
U7_GAA(
x4)
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)
U9_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_GGA(
x1,
x2,
x3,
x4,
x6)
LE_IN_GG(
x1,
x2) =
LE_IN_GG(
x1,
x2)
U14_GG(
x1,
x2,
x3) =
U14_GG(
x3)
U10_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_GGA(
x1,
x6)
U11_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_GGA(
x1,
x2,
x3,
x4,
x6)
GT_IN_GG(
x1,
x2) =
GT_IN_GG(
x1,
x2)
U13_GG(
x1,
x2,
x3) =
U13_GG(
x3)
U12_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_GGA(
x3,
x6)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 16 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(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
GT_IN_GG(
x1,
x2) =
GT_IN_GG(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(8) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(9) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(10) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(12) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
(13) 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(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
LE_IN_GG(
x1,
x2) =
LE_IN_GG(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(15) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(16) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(17) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(19) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
(20) TRUE
(21) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
MERGE_IN_GGA(
x1,
x2,
x3) =
MERGE_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_GGA(
x1,
x2,
x3,
x4,
x6)
U11_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_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:
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
MERGE_IN_GGA(
x1,
x2,
x3) =
MERGE_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_GGA(
x1,
x2,
x3,
x4,
x6)
U11_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_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:
U9_GGA(X, L1, Y, L2, le_out_gg) → MERGE_IN_GGA(L1, .(Y, L2))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, gt_out_gg) → MERGE_IN_GGA(.(X, L1), L2)
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)) → U13_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U14_gg(le_out_gg) → le_out_gg
U13_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)
U14_gg(x0)
U13_gg(x0)
We have to consider all (P,Q,R)-chains.
(26) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U9_GGA(X, L1, Y, L2, le_out_gg) → MERGE_IN_GGA(L1, .(Y, L2))
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(U11_GGA(x1, x2, x3, x4, x5)) = x1 + x2
POL(U13_gg(x1)) = 0
POL(U14_gg(x1)) = 1
POL(U9_GGA(x1, x2, x3, x4, x5)) = x2 + x5
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)) → U14_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U14_gg(le_out_gg) → le_out_gg
(27) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, gt_out_gg) → MERGE_IN_GGA(.(X, L1), L2)
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)) → U13_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U14_gg(le_out_gg) → le_out_gg
U13_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)
U14_gg(x0)
U13_gg(x0)
We have to consider all (P,Q,R)-chains.
(28) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(29) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, gt_out_gg) → MERGE_IN_GGA(.(X, L1), L2)
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)) → U13_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U14_gg(le_out_gg) → le_out_gg
U13_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)
U14_gg(x0)
U13_gg(x0)
We have to consider all (P,Q,R)-chains.
(30) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(31) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, gt_out_gg) → MERGE_IN_GGA(.(X, L1), L2)
The TRS R consists of the following rules:
gt_in_gg(s(X), s(Y)) → U13_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U13_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)
U14_gg(x0)
U13_gg(x0)
We have to consider all (P,Q,R)-chains.
(32) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
le_in_gg(x0, x1)
U14_gg(x0)
(33) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, gt_out_gg) → MERGE_IN_GGA(.(X, L1), L2)
The TRS R consists of the following rules:
gt_in_gg(s(X), s(Y)) → U13_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U13_gg(gt_out_gg) → gt_out_gg
The set Q consists of the following terms:
gt_in_gg(x0, x1)
U13_gg(x0)
We have to consider all (P,Q,R)-chains.
(34) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- U11_GGA(X, L1, Y, L2, gt_out_gg) → MERGE_IN_GGA(.(X, L1), L2)
The graph contains the following edges 4 >= 2
- MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
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:
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
SPLIT2_IN_GAA(
x1,
x2,
x3) =
SPLIT2_IN_GAA(
x1)
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:
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
SPLIT2_IN_GAA(
x1,
x2,
x3) =
SPLIT2_IN_GAA(
x1)
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:
SPLIT2_IN_GAA(.(X, .(Y, L1))) → SPLIT_IN_GAA(L1)
SPLIT_IN_GAA(L1) → SPLIT2_IN_GAA(L1)
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(L1) → SPLIT2_IN_GAA(L1)
The graph contains the following edges 1 >= 1
- SPLIT2_IN_GAA(.(X, .(Y, L1))) → SPLIT_IN_GAA(L1)
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(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x3,
x6)
MERGESORT_IN_GA(
x1,
x2) =
MERGESORT_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x5)
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GA(
x5,
x6)
We have to consider all (P,R,Pi)-chains
(44) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(45) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split2_out_gaa(L3, L4)) → U2_GA(L4, mergesort_in_ga(L3))
U2_GA(L4, mergesort_out_ga(L5)) → MERGESORT_IN_GA(L4)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split0_in_gaa([]) → split0_out_gaa([], [])
U5_gaa(split0_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
U7_gaa(split2_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
U1_ga(split2_out_gaa(L3, L4)) → U2_ga(L4, mergesort_in_ga(L3))
U2_ga(L4, mergesort_out_ga(L5)) → U3_ga(L5, mergesort_in_ga(L4))
U3_ga(L5, mergesort_out_ga(L6)) → U4_ga(merge_in_gga(L5, L6))
merge_in_gga([], L1) → merge_out_gga(L1)
merge_in_gga(L1, []) → merge_out_gga(L1)
merge_in_gga(.(X, L1), .(Y, L2)) → U9_gga(X, L1, Y, L2, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U14_gg(le_out_gg) → le_out_gg
U9_gga(X, L1, Y, L2, le_out_gg) → U10_gga(X, merge_in_gga(L1, .(Y, L2)))
merge_in_gga(.(X, L1), .(Y, L2)) → U11_gga(X, L1, Y, L2, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U13_gg(gt_out_gg) → gt_out_gg
U11_gga(X, L1, Y, L2, gt_out_gg) → U12_gga(Y, merge_in_gga(.(X, L1), L2))
U12_gga(Y, merge_out_gga(L3)) → merge_out_gga(.(Y, L3))
U10_gga(X, merge_out_gga(L3)) → merge_out_gga(.(X, L3))
U4_ga(merge_out_gga(L2)) → mergesort_out_ga(L2)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U7_gaa(x0)
U8_gaa(x0, x1, x2)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U14_gg(x0)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1)
U10_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
(46) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U1_GA(split2_out_gaa(L3, L4)) → U2_GA(L4, mergesort_in_ga(L3))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
| POL(split2_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)) = | 1 | + | | · | x1 |
| POL(.(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split2_in_gaa(x1)) = | | + | | · | x1 |
| POL(U8_gaa(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(split_in_gaa(x1)) = | | + | | · | x1 |
| POL(U2_ga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U3_ga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(merge_in_gga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split_out_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split0_in_gaa(x1)) = | | + | | · | x1 |
| POL(split1_in_gaa(x1)) = | | + | | · | x1 |
| POL(split0_out_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split1_out_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(merge_out_gga(x1)) = | | + | | · | x1 |
| POL(U9_gga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(le_in_gg(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U11_gga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(gt_in_gg(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U10_gga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U12_gga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
U1_ga(split2_out_gaa(L3, L4)) → U2_ga(L4, mergesort_in_ga(L3))
U2_ga(L4, mergesort_out_ga(L5)) → U3_ga(L5, mergesort_in_ga(L4))
U3_ga(L5, mergesort_out_ga(L6)) → U4_ga(merge_in_gga(L5, L6))
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
U7_gaa(split2_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
split0_in_gaa([]) → split0_out_gaa([], [])
U5_gaa(split0_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
U4_ga(merge_out_gga(L2)) → mergesort_out_ga(L2)
(47) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U2_GA(L4, mergesort_out_ga(L5)) → MERGESORT_IN_GA(L4)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split0_in_gaa([]) → split0_out_gaa([], [])
U5_gaa(split0_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
U7_gaa(split2_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
U1_ga(split2_out_gaa(L3, L4)) → U2_ga(L4, mergesort_in_ga(L3))
U2_ga(L4, mergesort_out_ga(L5)) → U3_ga(L5, mergesort_in_ga(L4))
U3_ga(L5, mergesort_out_ga(L6)) → U4_ga(merge_in_gga(L5, L6))
merge_in_gga([], L1) → merge_out_gga(L1)
merge_in_gga(L1, []) → merge_out_gga(L1)
merge_in_gga(.(X, L1), .(Y, L2)) → U9_gga(X, L1, Y, L2, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U14_gg(le_out_gg) → le_out_gg
U9_gga(X, L1, Y, L2, le_out_gg) → U10_gga(X, merge_in_gga(L1, .(Y, L2)))
merge_in_gga(.(X, L1), .(Y, L2)) → U11_gga(X, L1, Y, L2, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U13_gg(gt_out_gg) → gt_out_gg
U11_gga(X, L1, Y, L2, gt_out_gg) → U12_gga(Y, merge_in_gga(.(X, L1), L2))
U12_gga(Y, merge_out_gga(L3)) → merge_out_gga(.(Y, L3))
U10_gga(X, merge_out_gga(L3)) → merge_out_gga(.(X, L3))
U4_ga(merge_out_gga(L2)) → mergesort_out_ga(L2)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U7_gaa(x0)
U8_gaa(x0, x1, x2)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U14_gg(x0)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1)
U10_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
(48) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(49) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split0_in_gaa([]) → split0_out_gaa([], [])
U5_gaa(split0_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
U7_gaa(split2_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
U1_ga(split2_out_gaa(L3, L4)) → U2_ga(L4, mergesort_in_ga(L3))
U2_ga(L4, mergesort_out_ga(L5)) → U3_ga(L5, mergesort_in_ga(L4))
U3_ga(L5, mergesort_out_ga(L6)) → U4_ga(merge_in_gga(L5, L6))
merge_in_gga([], L1) → merge_out_gga(L1)
merge_in_gga(L1, []) → merge_out_gga(L1)
merge_in_gga(.(X, L1), .(Y, L2)) → U9_gga(X, L1, Y, L2, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U14_gg(le_out_gg) → le_out_gg
U9_gga(X, L1, Y, L2, le_out_gg) → U10_gga(X, merge_in_gga(L1, .(Y, L2)))
merge_in_gga(.(X, L1), .(Y, L2)) → U11_gga(X, L1, Y, L2, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U13_gg(gt_out_gg) → gt_out_gg
U11_gga(X, L1, Y, L2, gt_out_gg) → U12_gga(Y, merge_in_gga(.(X, L1), L2))
U12_gga(Y, merge_out_gga(L3)) → merge_out_gga(.(Y, L3))
U10_gga(X, merge_out_gga(L3)) → merge_out_gga(.(X, L3))
U4_ga(merge_out_gga(L2)) → mergesort_out_ga(L2)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U7_gaa(x0)
U8_gaa(x0, x1, x2)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U14_gg(x0)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1)
U10_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
(50) 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.
(51) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
The TRS R consists of the following rules:
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
U7_gaa(split2_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split0_in_gaa([]) → split0_out_gaa([], [])
U5_gaa(split0_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U7_gaa(x0)
U8_gaa(x0, x1, x2)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U14_gg(x0)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1)
U10_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
(52) 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)
U14_gg(x0)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1)
U10_gga(x0, x1)
U4_ga(x0)
(53) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
The TRS R consists of the following rules:
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
U7_gaa(split2_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split0_in_gaa([]) → split0_out_gaa([], [])
U5_gaa(split0_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
The set Q consists of the following terms:
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U7_gaa(x0)
U8_gaa(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(54) 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 rules of the TRS R:
U7_gaa(split2_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
split0_in_gaa([]) → split0_out_gaa([], [])
Used ordering: Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 2 + x1 + 2·x2
POL(MERGESORT_IN_GA(x1)) = 2·x1
POL(U1_GA(x1)) = 2·x1
POL(U5_gaa(x1)) = x1
POL(U6_gaa(x1)) = 1 + x1
POL(U7_gaa(x1)) = 2 + x1
POL(U8_gaa(x1, x2, x3)) = 2 + x1 + x2 + 2·x3
POL([]) = 1
POL(split0_in_gaa(x1)) = 2 + 2·x1
POL(split0_out_gaa(x1, x2)) = 1 + x1 + x2
POL(split1_in_gaa(x1)) = 1 + 2·x1
POL(split1_out_gaa(x1, x2)) = 2·x1 + x2
POL(split2_in_gaa(x1)) = x1
POL(split2_out_gaa(x1, x2)) = x1 + x2
POL(split_in_gaa(x1)) = 2 + 2·x1
POL(split_out_gaa(x1, x2)) = 1 + x1 + x2
(55) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
The TRS R consists of the following rules:
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
U5_gaa(split0_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
The set Q consists of the following terms:
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U7_gaa(x0)
U8_gaa(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(56) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(57) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
The TRS R consists of the following rules:
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
The set Q consists of the following terms:
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U7_gaa(x0)
U8_gaa(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(58) 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 rules of the TRS R:
split_in_gaa(L1) → U5_gaa(split0_in_gaa(L1))
split_in_gaa(L1) → U7_gaa(split2_in_gaa(L1))
Used ordering: Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 1 + x1 + x2
POL(MERGESORT_IN_GA(x1)) = x1
POL(U1_GA(x1)) = x1
POL(U5_gaa(x1)) = x1
POL(U6_gaa(x1)) = x1
POL(U7_gaa(x1)) = x1
POL(U8_gaa(x1, x2, x3)) = 1 + x1 + x2 + x3
POL([]) = 0
POL(split0_in_gaa(x1)) = x1
POL(split1_in_gaa(x1)) = 1 + x1
POL(split1_out_gaa(x1, x2)) = 1 + x1 + x2
POL(split2_in_gaa(x1)) = x1
POL(split2_out_gaa(x1, x2)) = x1 + x2
POL(split_in_gaa(x1)) = 1 + x1
POL(split_out_gaa(x1, x2)) = 1 + x1 + x2
(59) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
The TRS R consists of the following rules:
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
The set Q consists of the following terms:
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U7_gaa(x0)
U8_gaa(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(60) 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].
split0_in_gaa(x0)
U5_gaa(x0)
U7_gaa(x0)
(61) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
The TRS R consists of the following rules:
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
The set Q consists of the following terms:
split2_in_gaa(x0)
split_in_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U8_gaa(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(62) 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:
U1_GA(split2_out_gaa(L3, L4)) → MERGESORT_IN_GA(L3)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(split2_in_gaa(.(X, .(Y, L1))))
Strictly oriented rules of the TRS R:
U8_gaa(X, Y, split_out_gaa(L2, L3)) → split2_out_gaa(.(X, L2), .(Y, L3))
Used ordering: Polynomial interpretation [POLO]:
POL(.(x1, x2)) = x1 + 2·x2
POL(MERGESORT_IN_GA(x1)) = 1 + 2·x1
POL(U1_GA(x1)) = 2·x1
POL(U6_gaa(x1)) = x1
POL(U8_gaa(x1, x2, x3)) = x1 + 2·x2 + 2·x3
POL([]) = 2
POL(split1_in_gaa(x1)) = 2·x1
POL(split1_out_gaa(x1, x2)) = 2 + x1 + x2
POL(split2_in_gaa(x1)) = x1
POL(split2_out_gaa(x1, x2)) = 1 + x1 + x2
POL(split_in_gaa(x1)) = 2·x1
POL(split_out_gaa(x1, x2)) = 2 + x1 + x2
(63) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, split_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(split1_in_gaa(L1))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), [])
U6_gaa(split1_out_gaa(L2, L3)) → split_out_gaa(L2, L3)
The set Q consists of the following terms:
split2_in_gaa(x0)
split_in_gaa(x0)
split1_in_gaa(x0)
U6_gaa(x0)
U8_gaa(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(64) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(65) TRUE
(66) 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)
split2_in: (b,f,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(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x1,
x2,
x3,
x4,
x6)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(67) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x1,
x2,
x3,
x4,
x6)
(68) 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(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4)
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT0_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT1_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → MERGE_IN_GGA(L5, L6, L2)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U14_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_GGA(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U13_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_GGA(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_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)
SPLIT2_IN_GAA(
x1,
x2,
x3) =
SPLIT2_IN_GAA(
x1)
U8_GAA(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_GAA(
x1,
x2,
x3,
x6)
SPLIT_IN_GAA(
x1,
x2,
x3) =
SPLIT_IN_GAA(
x1)
U5_GAA(
x1,
x2,
x3,
x4) =
U5_GAA(
x1,
x4)
SPLIT0_IN_GAA(
x1,
x2,
x3) =
SPLIT0_IN_GAA(
x1)
U6_GAA(
x1,
x2,
x3,
x4) =
U6_GAA(
x1,
x4)
SPLIT1_IN_GAA(
x1,
x2,
x3) =
SPLIT1_IN_GAA(
x1)
U7_GAA(
x1,
x2,
x3,
x4) =
U7_GAA(
x1,
x4)
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)
U9_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_GGA(
x1,
x2,
x3,
x4,
x6)
LE_IN_GG(
x1,
x2) =
LE_IN_GG(
x1,
x2)
U14_GG(
x1,
x2,
x3) =
U14_GG(
x1,
x2,
x3)
U10_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_GGA(
x1,
x2,
x3,
x4,
x6)
U11_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_GGA(
x1,
x2,
x3,
x4,
x6)
GT_IN_GG(
x1,
x2) =
GT_IN_GG(
x1,
x2)
U13_GG(
x1,
x2,
x3) =
U13_GG(
x1,
x2,
x3)
U12_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_GGA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(69) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4)
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT0_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT1_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → MERGE_IN_GGA(L5, L6, L2)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U14_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_GGA(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U13_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_GGA(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_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)
SPLIT2_IN_GAA(
x1,
x2,
x3) =
SPLIT2_IN_GAA(
x1)
U8_GAA(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_GAA(
x1,
x2,
x3,
x6)
SPLIT_IN_GAA(
x1,
x2,
x3) =
SPLIT_IN_GAA(
x1)
U5_GAA(
x1,
x2,
x3,
x4) =
U5_GAA(
x1,
x4)
SPLIT0_IN_GAA(
x1,
x2,
x3) =
SPLIT0_IN_GAA(
x1)
U6_GAA(
x1,
x2,
x3,
x4) =
U6_GAA(
x1,
x4)
SPLIT1_IN_GAA(
x1,
x2,
x3) =
SPLIT1_IN_GAA(
x1)
U7_GAA(
x1,
x2,
x3,
x4) =
U7_GAA(
x1,
x4)
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)
U9_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_GGA(
x1,
x2,
x3,
x4,
x6)
LE_IN_GG(
x1,
x2) =
LE_IN_GG(
x1,
x2)
U14_GG(
x1,
x2,
x3) =
U14_GG(
x1,
x2,
x3)
U10_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_GGA(
x1,
x2,
x3,
x4,
x6)
U11_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_GGA(
x1,
x2,
x3,
x4,
x6)
GT_IN_GG(
x1,
x2) =
GT_IN_GG(
x1,
x2)
U13_GG(
x1,
x2,
x3) =
U13_GG(
x1,
x2,
x3)
U12_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_GGA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(70) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 16 less nodes.
(71) Complex Obligation (AND)
(72) 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(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_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
(73) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(74) 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
(75) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(76) 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.
(77) 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
(78) TRUE
(79) 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(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_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
(80) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(81) 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
(82) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(83) 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.
(84) 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
(85) TRUE
(86) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x1,
x2,
x3,
x4,
x6)
MERGE_IN_GGA(
x1,
x2,
x3) =
MERGE_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_GGA(
x1,
x2,
x3,
x4,
x6)
U11_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_GGA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(87) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(88) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(
x1,
x2,
x3) =
U14_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)
U13_gg(
x1,
x2,
x3) =
U13_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)
U9_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_GGA(
x1,
x2,
x3,
x4,
x6)
U11_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_GGA(
x1,
x2,
x3,
x4,
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:
U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2)
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(91) 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:
U11_GGA(X, L1, Y, L2, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2)
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(U11_GGA(x1, x2, x3, x4, x5)) = 2·x1 + 2·x2 + 2·x3 + 2·x4 + x5
POL(U13_gg(x1, x2, x3)) = 2·x1 + 2·x2 + x3
POL(U14_gg(x1, x2, x3)) = x1 + x2 + x3
POL(U9_GGA(x1, x2, x3, x4, x5)) = x1 + 2·x2 + 2·x3 + 2·x4 + 2·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
(92) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)) → U13_gg(X, Y, gt_in_gg(X, Y))
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
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:
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)) → U13_gg(X, Y, gt_in_gg(X, Y))
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
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:
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)
U14_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)
U14_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
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].
gt_in_gg(x0, x1)
U13_gg(x0, x1, x2)
(98) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
The TRS R consists of the following rules:
le_in_gg(s(X), s(Y)) → U14_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)
U14_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)
U14_gg(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(99) 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:
- U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
The graph contains the following edges 2 >= 1
- MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4
(100) TRUE
(101) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_gga(
x1,
x2,
x3,
x4,
x6)
SPLIT2_IN_GAA(
x1,
x2,
x3) =
SPLIT2_IN_GAA(
x1)
SPLIT_IN_GAA(
x1,
x2,
x3) =
SPLIT_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(102) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(103) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
SPLIT2_IN_GAA(
x1,
x2,
x3) =
SPLIT2_IN_GAA(
x1)
SPLIT_IN_GAA(
x1,
x2,
x3) =
SPLIT_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(104) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(105) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SPLIT2_IN_GAA(.(X, .(Y, L1))) → SPLIT_IN_GAA(L1)
SPLIT_IN_GAA(L1) → SPLIT2_IN_GAA(L1)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(106) 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(L1) → SPLIT2_IN_GAA(L1)
The graph contains the following edges 1 >= 1
- SPLIT2_IN_GAA(.(X, .(Y, L1))) → SPLIT_IN_GAA(L1)
The graph contains the following edges 1 > 1
(107) TRUE
(108) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
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)
split2_in_gaa(
x1,
x2,
x3) =
split2_in_gaa(
x1)
U8_gaa(
x1,
x2,
x3,
x4,
x5,
x6) =
U8_gaa(
x1,
x2,
x3,
x6)
split_in_gaa(
x1,
x2,
x3) =
split_in_gaa(
x1)
U5_gaa(
x1,
x2,
x3,
x4) =
U5_gaa(
x1,
x4)
split0_in_gaa(
x1,
x2,
x3) =
split0_in_gaa(
x1)
split0_out_gaa(
x1,
x2,
x3) =
split0_out_gaa(
x1,
x2,
x3)
split_out_gaa(
x1,
x2,
x3) =
split_out_gaa(
x1,
x2,
x3)
U6_gaa(
x1,
x2,
x3,
x4) =
U6_gaa(
x1,
x4)
split1_in_gaa(
x1,
x2,
x3) =
split1_in_gaa(
x1)
split1_out_gaa(
x1,
x2,
x3) =
split1_out_gaa(
x1,
x2,
x3)
U7_gaa(
x1,
x2,
x3,
x4) =
U7_gaa(
x1,
x4)
split2_out_gaa(
x1,
x2,
x3) =
split2_out_gaa(
x1,
x2,
x3)
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)
U9_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U9_gga(
x1,
x2,
x3,
x4,
x6)
le_in_gg(
x1,
x2) =
le_in_gg(
x1,
x2)
s(
x1) =
s(
x1)
U14_gg(
x1,
x2,
x3) =
U14_gg(
x1,
x2,
x3)
0 =
0
le_out_gg(
x1,
x2) =
le_out_gg(
x1,
x2)
U10_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U10_gga(
x1,
x2,
x3,
x4,
x6)
U11_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U11_gga(
x1,
x2,
x3,
x4,
x6)
gt_in_gg(
x1,
x2) =
gt_in_gg(
x1,
x2)
U13_gg(
x1,
x2,
x3) =
U13_gg(
x1,
x2,
x3)
gt_out_gg(
x1,
x2) =
gt_out_gg(
x1,
x2)
U12_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U12_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
(109) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(110) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L4, mergesort_in_ga(L3))
U2_GA(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))
U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3)
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, L1, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(L1, split0_in_gaa(L1))
split0_in_gaa([]) → split0_out_gaa([], [], [])
U5_gaa(L1, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U6_gaa(L1, split1_in_gaa(L1))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U7_gaa(L1, split2_in_gaa(L1))
U7_gaa(L1, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L4, mergesort_in_ga(L3))
U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L5, mergesort_in_ga(L4))
U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, merge_in_gga(L5, L6))
merge_in_gga([], L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, []) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2)) → U9_gga(X, L1, Y, L2, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, merge_in_gga(L1, .(Y, L2)))
merge_in_gga(.(X, L1), .(Y, L2)) → U11_gga(X, L1, Y, L2, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, merge_in_gga(.(X, L1), L2))
U12_gga(X, L1, Y, L2, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0, x1)
split1_in_gaa(x0)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U8_gaa(x0, x1, x2, x3)
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)
U14_gg(x0, x1, x2)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0, x1, x2)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1, x2, x3, x4)
U10_gga(x0, x1, x2, x3, x4)
U4_ga(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(111) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
| POL(U1_GA(x1, x2, x3, x4)) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(split2_out_gaa(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(.(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U2_GA(x1, x2, x3, x4, x5)) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(mergesort_in_ga(x1)) = | | + | | · | x1 |
| POL(mergesort_out_ga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(MERGESORT_IN_GA(x1)) = | 1 | + | | · | x1 |
| POL(split2_in_gaa(x1)) = | | + | | · | x1 |
| POL(U1_ga(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(U8_gaa(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(split_in_gaa(x1)) = | | + | | · | x1 |
| POL(U2_ga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(U3_ga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(U4_ga(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(merge_in_gga(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U7_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split_out_gaa(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(U5_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split0_in_gaa(x1)) = | | + | | · | x1 |
| POL(U6_gaa(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(split1_in_gaa(x1)) = | | + | | · | x1 |
| POL(split0_out_gaa(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(split1_out_gaa(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(merge_out_gga(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(U9_gga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(le_in_gg(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U11_gga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(gt_in_gg(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U14_gg(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(le_out_gg(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U10_gga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
| POL(U13_gg(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(gt_out_gg(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(U12_gga(x1, x2, x3, x4, x5)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
The following usable rules [FROCOS05] were oriented:
mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, L1, split_in_gaa(L1))
U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L4, mergesort_in_ga(L3))
U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L5, mergesort_in_ga(L4))
U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, merge_in_gga(L5, L6))
split_in_gaa(L1) → U7_gaa(L1, split2_in_gaa(L1))
U7_gaa(L1, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U5_gaa(L1, split0_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(L1, split1_in_gaa(L1))
U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
split0_in_gaa([]) → split0_out_gaa([], [], [])
U5_gaa(L1, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
(112) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L4, mergesort_in_ga(L3))
U2_GA(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, L1, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(L1, split0_in_gaa(L1))
split0_in_gaa([]) → split0_out_gaa([], [], [])
U5_gaa(L1, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U6_gaa(L1, split1_in_gaa(L1))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U7_gaa(L1, split2_in_gaa(L1))
U7_gaa(L1, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L4, mergesort_in_ga(L3))
U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L5, mergesort_in_ga(L4))
U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, merge_in_gga(L5, L6))
merge_in_gga([], L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, []) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2)) → U9_gga(X, L1, Y, L2, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_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)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, merge_in_gga(L1, .(Y, L2)))
merge_in_gga(.(X, L1), .(Y, L2)) → U11_gga(X, L1, Y, L2, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, merge_in_gga(.(X, L1), L2))
U12_gga(X, L1, Y, L2, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0, x1)
split1_in_gaa(x0)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U8_gaa(x0, x1, x2, x3)
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)
U14_gg(x0, x1, x2)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0, x1, x2)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1, x2, x3, x4)
U10_gga(x0, x1, x2, x3, x4)
U4_ga(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.