11.40/3.85 YES 11.84/3.90 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 11.84/3.90 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.84/3.90 11.84/3.90 11.84/3.90 Left Termination of the query pattern 11.84/3.90 11.84/3.90 mergesort(g,a) 11.84/3.90 11.84/3.90 w.r.t. the given Prolog program could successfully be proven: 11.84/3.90 11.84/3.90 (0) Prolog 11.84/3.90 (1) PrologToPiTRSProof [SOUND, 35 ms] 11.84/3.90 (2) PiTRS 11.84/3.90 (3) DependencyPairsProof [EQUIVALENT, 34 ms] 11.84/3.90 (4) PiDP 11.84/3.90 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 11.84/3.90 (6) AND 11.84/3.90 (7) PiDP 11.84/3.90 (8) UsableRulesProof [EQUIVALENT, 0 ms] 11.84/3.90 (9) PiDP 11.84/3.90 (10) PiDPToQDPProof [EQUIVALENT, 0 ms] 11.84/3.90 (11) QDP 11.84/3.90 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.84/3.90 (13) YES 11.84/3.90 (14) PiDP 11.84/3.90 (15) UsableRulesProof [EQUIVALENT, 0 ms] 11.84/3.90 (16) PiDP 11.84/3.90 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 11.84/3.90 (18) QDP 11.84/3.90 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.84/3.90 (20) YES 11.84/3.90 (21) PiDP 11.84/3.90 (22) UsableRulesProof [EQUIVALENT, 0 ms] 11.84/3.90 (23) PiDP 11.84/3.90 (24) PiDPToQDPProof [SOUND, 0 ms] 11.84/3.90 (25) QDP 11.84/3.90 (26) MRRProof [EQUIVALENT, 46 ms] 11.84/3.90 (27) QDP 11.84/3.90 (28) DependencyGraphProof [EQUIVALENT, 0 ms] 11.84/3.90 (29) QDP 11.84/3.90 (30) UsableRulesProof [EQUIVALENT, 0 ms] 11.84/3.90 (31) QDP 11.84/3.90 (32) QReductionProof [EQUIVALENT, 0 ms] 11.84/3.90 (33) QDP 11.84/3.90 (34) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.84/3.90 (35) YES 11.84/3.90 (36) PiDP 11.84/3.90 (37) UsableRulesProof [EQUIVALENT, 0 ms] 11.84/3.90 (38) PiDP 11.84/3.90 (39) PiDPToQDPProof [SOUND, 0 ms] 11.84/3.90 (40) QDP 11.84/3.90 (41) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.84/3.90 (42) YES 11.84/3.90 (43) PiDP 11.84/3.90 (44) PiDPToQDPProof [SOUND, 3 ms] 11.84/3.90 (45) QDP 11.84/3.90 (46) QDPOrderProof [EQUIVALENT, 371 ms] 11.84/3.90 (47) QDP 11.84/3.90 (48) DependencyGraphProof [EQUIVALENT, 0 ms] 11.84/3.90 (49) TRUE 11.84/3.90 11.84/3.90 11.84/3.90 ---------------------------------------- 11.84/3.90 11.84/3.90 (0) 11.84/3.90 Obligation: 11.84/3.90 Clauses: 11.84/3.90 11.84/3.90 mergesort([], []). 11.84/3.90 mergesort(.(X, []), .(X, [])). 11.84/3.90 mergesort(.(X, .(Y, L1)), L2) :- ','(split2(.(X, .(Y, L1)), L3, L4), ','(mergesort(L3, L5), ','(mergesort(L4, L6), merge(L5, L6, L2)))). 11.84/3.90 split(L1, L2, L3) :- split0(L1, L2, L3). 11.84/3.90 split(L1, L2, L3) :- split1(L1, L2, L3). 11.84/3.90 split(L1, L2, L3) :- split2(L1, L2, L3). 11.84/3.90 split0([], [], []). 11.84/3.90 split1(.(X, []), .(X, []), []). 11.84/3.90 split2(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) :- split(L1, L2, L3). 11.84/3.90 merge([], L1, L1). 11.84/3.90 merge(L1, [], L1). 11.84/3.90 merge(.(X, L1), .(Y, L2), .(X, L3)) :- ','(le(X, Y), merge(L1, .(Y, L2), L3)). 11.84/3.90 merge(.(X, L1), .(Y, L2), .(Y, L3)) :- ','(gt(X, Y), merge(.(X, L1), L2, L3)). 11.84/3.90 gt(s(X), s(Y)) :- gt(X, Y). 11.84/3.90 gt(s(X), 0). 11.84/3.90 le(s(X), s(Y)) :- le(X, Y). 11.84/3.90 le(0, s(Y)). 11.84/3.90 le(0, 0). 11.84/3.90 11.84/3.90 11.84/3.90 Query: mergesort(g,a) 11.84/3.90 ---------------------------------------- 11.84/3.90 11.84/3.90 (1) PrologToPiTRSProof (SOUND) 11.84/3.90 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 11.84/3.90 11.84/3.90 mergesort_in_2: (b,f) 11.84/3.90 11.84/3.90 split2_in_3: (b,f,f) 11.84/3.90 11.84/3.90 split_in_3: (b,f,f) 11.84/3.90 11.84/3.90 merge_in_3: (b,b,f) 11.84/3.90 11.84/3.90 le_in_2: (b,b) 11.84/3.90 11.84/3.90 gt_in_2: (b,b) 11.84/3.90 11.84/3.90 Transforming Prolog into the following Term Rewriting System: 11.84/3.90 11.84/3.90 Pi-finite rewrite system: 11.84/3.90 The TRS R consists of the following rules: 11.84/3.90 11.84/3.90 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.90 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.90 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.90 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.90 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.90 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.90 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.90 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.90 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.90 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.90 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.90 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.90 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.90 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)) 11.84/3.90 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.90 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.90 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.90 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.90 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.90 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.90 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.90 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.90 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.90 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)) 11.84/3.90 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.90 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.90 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.90 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.90 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)) 11.84/3.90 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.90 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.90 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.90 11.84/3.90 The argument filtering Pi contains the following mapping: 11.84/3.90 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.90 11.84/3.90 [] = [] 11.84/3.90 11.84/3.90 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.90 11.84/3.90 .(x1, x2) = .(x1, x2) 11.84/3.90 11.84/3.90 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.90 11.84/3.90 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.90 11.84/3.90 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.90 11.84/3.90 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.90 11.84/3.90 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.90 11.84/3.90 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.90 11.84/3.90 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.90 11.84/3.90 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.90 11.84/3.90 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.90 11.84/3.90 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.90 11.84/3.90 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.90 11.84/3.90 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.90 11.84/3.90 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.90 11.84/3.90 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.90 11.84/3.90 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.90 11.84/3.90 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.90 11.84/3.90 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.90 11.84/3.90 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.90 11.84/3.90 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.90 11.84/3.90 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.90 11.84/3.90 s(x1) = s(x1) 11.84/3.90 11.84/3.90 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.90 11.84/3.90 0 = 0 11.84/3.90 11.84/3.90 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.90 11.84/3.90 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.90 11.84/3.90 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.90 11.84/3.90 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.90 11.84/3.90 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.90 11.84/3.90 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.90 11.84/3.90 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.90 11.84/3.90 11.84/3.90 11.84/3.90 11.84/3.90 11.84/3.90 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 11.84/3.90 11.84/3.90 11.84/3.90 11.84/3.90 ---------------------------------------- 11.84/3.90 11.84/3.90 (2) 11.84/3.90 Obligation: 11.84/3.90 Pi-finite rewrite system: 11.84/3.90 The TRS R consists of the following rules: 11.84/3.90 11.84/3.90 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.90 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.90 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.90 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.90 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.90 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.90 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.90 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.90 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.90 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.90 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.90 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.90 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.90 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)) 11.84/3.90 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.90 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.90 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.90 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.90 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.90 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.90 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.90 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.90 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.90 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)) 11.84/3.90 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.90 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.90 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.90 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.90 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)) 11.84/3.90 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.90 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.90 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.90 11.84/3.90 The argument filtering Pi contains the following mapping: 11.84/3.90 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.90 11.84/3.90 [] = [] 11.84/3.90 11.84/3.90 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.90 11.84/3.90 .(x1, x2) = .(x1, x2) 11.84/3.90 11.84/3.90 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.90 11.84/3.90 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.90 11.84/3.90 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.90 11.84/3.90 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.90 11.84/3.90 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.90 11.84/3.90 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.90 11.84/3.90 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.90 11.84/3.90 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.90 11.84/3.90 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.90 11.84/3.90 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.90 11.84/3.90 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.90 11.84/3.90 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.90 11.84/3.90 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.90 11.84/3.90 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.90 11.84/3.90 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.90 11.84/3.90 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.90 11.84/3.90 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.90 11.84/3.90 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.90 11.84/3.90 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.90 11.84/3.90 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.90 11.84/3.90 s(x1) = s(x1) 11.84/3.90 11.84/3.90 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.90 11.84/3.90 0 = 0 11.84/3.90 11.84/3.90 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.90 11.84/3.90 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.90 11.84/3.90 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.90 11.84/3.90 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.90 11.84/3.90 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.90 11.84/3.90 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.90 11.84/3.90 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.90 11.84/3.90 11.84/3.90 11.84/3.90 ---------------------------------------- 11.84/3.90 11.84/3.90 (3) DependencyPairsProof (EQUIVALENT) 11.84/3.90 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 11.84/3.90 Pi DP problem: 11.84/3.90 The TRS P consists of the following rules: 11.84/3.90 11.84/3.90 MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.90 MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4) 11.84/3.90 SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.90 SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> SPLIT_IN_GAA(L1, L2, L3) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> SPLIT0_IN_GAA(L1, L2, L3) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> SPLIT1_IN_GAA(L1, L2, L3) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> SPLIT2_IN_GAA(L1, L2, L3) 11.84/3.91 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)) 11.84/3.91 U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3, L5) 11.84/3.91 U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4, L6) 11.84/3.91 U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> MERGE_IN_GGA(L5, L6, L2) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> LE_IN_GG(X, Y) 11.84/3.91 LE_IN_GG(s(X), s(Y)) -> U14_GG(X, Y, le_in_gg(X, Y)) 11.84/3.91 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 11.84/3.91 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)) 11.84/3.91 U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2), L3) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> GT_IN_GG(X, Y) 11.84/3.91 GT_IN_GG(s(X), s(Y)) -> U13_GG(X, Y, gt_in_gg(X, Y)) 11.84/3.91 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 11.84/3.91 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)) 11.84/3.91 U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2, L3) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 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)) 11.84/3.91 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.91 11.84/3.91 [] = [] 11.84/3.91 11.84/3.91 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.91 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.91 11.84/3.91 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.91 11.84/3.91 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.91 11.84/3.91 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.91 11.84/3.91 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.91 11.84/3.91 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.91 11.84/3.91 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.91 11.84/3.91 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.91 11.84/3.91 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.91 11.84/3.91 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.91 11.84/3.91 s(x1) = s(x1) 11.84/3.91 11.84/3.91 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 0 = 0 11.84/3.91 11.84/3.91 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.91 11.84/3.91 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1) 11.84/3.91 11.84/3.91 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1) 11.84/3.91 11.84/3.91 U8_GAA(x1, x2, x3, x4, x5, x6) = U8_GAA(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1) 11.84/3.91 11.84/3.91 U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x4) 11.84/3.91 11.84/3.91 SPLIT0_IN_GAA(x1, x2, x3) = SPLIT0_IN_GAA(x1) 11.84/3.91 11.84/3.91 U6_GAA(x1, x2, x3, x4) = U6_GAA(x1, x4) 11.84/3.91 11.84/3.91 SPLIT1_IN_GAA(x1, x2, x3) = SPLIT1_IN_GAA(x1) 11.84/3.91 11.84/3.91 U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4) 11.84/3.91 11.84/3.91 U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) 11.84/3.91 11.84/3.91 U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 11.84/3.91 11.84/3.91 U14_GG(x1, x2, x3) = U14_GG(x1, x2, x3) 11.84/3.91 11.84/3.91 U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 11.84/3.91 11.84/3.91 U13_GG(x1, x2, x3) = U13_GG(x1, x2, x3) 11.84/3.91 11.84/3.91 U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (4) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4) 11.84/3.91 SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.91 SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> SPLIT_IN_GAA(L1, L2, L3) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> SPLIT0_IN_GAA(L1, L2, L3) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> SPLIT1_IN_GAA(L1, L2, L3) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> SPLIT2_IN_GAA(L1, L2, L3) 11.84/3.91 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)) 11.84/3.91 U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3, L5) 11.84/3.91 U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4, L6) 11.84/3.91 U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> MERGE_IN_GGA(L5, L6, L2) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> LE_IN_GG(X, Y) 11.84/3.91 LE_IN_GG(s(X), s(Y)) -> U14_GG(X, Y, le_in_gg(X, Y)) 11.84/3.91 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 11.84/3.91 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)) 11.84/3.91 U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2), L3) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> GT_IN_GG(X, Y) 11.84/3.91 GT_IN_GG(s(X), s(Y)) -> U13_GG(X, Y, gt_in_gg(X, Y)) 11.84/3.91 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 11.84/3.91 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)) 11.84/3.91 U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2, L3) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 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)) 11.84/3.91 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.91 11.84/3.91 [] = [] 11.84/3.91 11.84/3.91 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.91 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.91 11.84/3.91 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.91 11.84/3.91 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.91 11.84/3.91 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.91 11.84/3.91 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.91 11.84/3.91 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.91 11.84/3.91 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.91 11.84/3.91 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.91 11.84/3.91 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.91 11.84/3.91 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.91 11.84/3.91 s(x1) = s(x1) 11.84/3.91 11.84/3.91 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 0 = 0 11.84/3.91 11.84/3.91 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.91 11.84/3.91 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1) 11.84/3.91 11.84/3.91 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1) 11.84/3.91 11.84/3.91 U8_GAA(x1, x2, x3, x4, x5, x6) = U8_GAA(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1) 11.84/3.91 11.84/3.91 U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x4) 11.84/3.91 11.84/3.91 SPLIT0_IN_GAA(x1, x2, x3) = SPLIT0_IN_GAA(x1) 11.84/3.91 11.84/3.91 U6_GAA(x1, x2, x3, x4) = U6_GAA(x1, x4) 11.84/3.91 11.84/3.91 SPLIT1_IN_GAA(x1, x2, x3) = SPLIT1_IN_GAA(x1) 11.84/3.91 11.84/3.91 U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4) 11.84/3.91 11.84/3.91 U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) 11.84/3.91 11.84/3.91 U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 11.84/3.91 11.84/3.91 U14_GG(x1, x2, x3) = U14_GG(x1, x2, x3) 11.84/3.91 11.84/3.91 U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 11.84/3.91 11.84/3.91 U13_GG(x1, x2, x3) = U13_GG(x1, x2, x3) 11.84/3.91 11.84/3.91 U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (5) DependencyGraphProof (EQUIVALENT) 11.84/3.91 The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 16 less nodes. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (6) 11.84/3.91 Complex Obligation (AND) 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (7) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 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)) 11.84/3.91 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.91 11.84/3.91 [] = [] 11.84/3.91 11.84/3.91 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.91 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.91 11.84/3.91 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.91 11.84/3.91 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.91 11.84/3.91 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.91 11.84/3.91 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.91 11.84/3.91 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.91 11.84/3.91 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.91 11.84/3.91 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.91 11.84/3.91 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.91 11.84/3.91 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.91 11.84/3.91 s(x1) = s(x1) 11.84/3.91 11.84/3.91 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 0 = 0 11.84/3.91 11.84/3.91 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.91 11.84/3.91 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (8) UsableRulesProof (EQUIVALENT) 11.84/3.91 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (9) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 11.84/3.91 11.84/3.91 R is empty. 11.84/3.91 Pi is empty. 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (10) PiDPToQDPProof (EQUIVALENT) 11.84/3.91 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (11) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 11.84/3.91 11.84/3.91 R is empty. 11.84/3.91 Q is empty. 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (12) QDPSizeChangeProof (EQUIVALENT) 11.84/3.91 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. 11.84/3.91 11.84/3.91 From the DPs we obtained the following set of size-change graphs: 11.84/3.91 *GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 11.84/3.91 The graph contains the following edges 1 > 1, 2 > 2 11.84/3.91 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (13) 11.84/3.91 YES 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (14) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 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)) 11.84/3.91 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.91 11.84/3.91 [] = [] 11.84/3.91 11.84/3.91 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.91 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.91 11.84/3.91 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.91 11.84/3.91 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.91 11.84/3.91 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.91 11.84/3.91 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.91 11.84/3.91 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.91 11.84/3.91 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.91 11.84/3.91 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.91 11.84/3.91 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.91 11.84/3.91 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.91 11.84/3.91 s(x1) = s(x1) 11.84/3.91 11.84/3.91 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 0 = 0 11.84/3.91 11.84/3.91 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.91 11.84/3.91 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (15) UsableRulesProof (EQUIVALENT) 11.84/3.91 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (16) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 11.84/3.91 11.84/3.91 R is empty. 11.84/3.91 Pi is empty. 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (17) PiDPToQDPProof (EQUIVALENT) 11.84/3.91 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (18) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 11.84/3.91 11.84/3.91 R is empty. 11.84/3.91 Q is empty. 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (19) QDPSizeChangeProof (EQUIVALENT) 11.84/3.91 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. 11.84/3.91 11.84/3.91 From the DPs we obtained the following set of size-change graphs: 11.84/3.91 *LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 11.84/3.91 The graph contains the following edges 1 > 1, 2 > 2 11.84/3.91 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (20) 11.84/3.91 YES 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (21) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2), L3) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2, L3) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 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)) 11.84/3.91 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.91 11.84/3.91 [] = [] 11.84/3.91 11.84/3.91 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.91 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.91 11.84/3.91 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.91 11.84/3.91 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.91 11.84/3.91 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.91 11.84/3.91 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.91 11.84/3.91 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.91 11.84/3.91 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.91 11.84/3.91 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.91 11.84/3.91 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.91 11.84/3.91 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.91 11.84/3.91 s(x1) = s(x1) 11.84/3.91 11.84/3.91 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 0 = 0 11.84/3.91 11.84/3.91 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.91 11.84/3.91 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) 11.84/3.91 11.84/3.91 U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (22) UsableRulesProof (EQUIVALENT) 11.84/3.91 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (23) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2), L3) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2, L3) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.91 11.84/3.91 s(x1) = s(x1) 11.84/3.91 11.84/3.91 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 0 = 0 11.84/3.91 11.84/3.91 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.91 11.84/3.91 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.91 11.84/3.91 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.91 11.84/3.91 MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) 11.84/3.91 11.84/3.91 U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (24) PiDPToQDPProof (SOUND) 11.84/3.91 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (25) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y)) 11.84/3.91 U11_GGA(X, L1, Y, L2, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 11.84/3.91 The set Q consists of the following terms: 11.84/3.91 11.84/3.91 le_in_gg(x0, x1) 11.84/3.91 gt_in_gg(x0, x1) 11.84/3.91 U14_gg(x0, x1, x2) 11.84/3.91 U13_gg(x0, x1, x2) 11.84/3.91 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (26) MRRProof (EQUIVALENT) 11.84/3.91 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. 11.84/3.91 11.84/3.91 Strictly oriented dependency pairs: 11.84/3.91 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y)) 11.84/3.91 11.84/3.91 Strictly oriented rules of the TRS R: 11.84/3.91 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 11.84/3.91 Used ordering: Polynomial interpretation [POLO]: 11.84/3.91 11.84/3.91 POL(.(x_1, x_2)) = 1 + 2*x_1 + x_2 11.84/3.91 POL(0) = 1 11.84/3.91 POL(MERGE_IN_GGA(x_1, x_2)) = 2*x_1 + 2*x_2 11.84/3.91 POL(U11_GGA(x_1, x_2, x_3, x_4, x_5)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 + 2*x_4 + 2*x_5 11.84/3.91 POL(U13_gg(x_1, x_2, x_3)) = x_1 + x_2 + x_3 11.84/3.91 POL(U14_gg(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 11.84/3.91 POL(U9_GGA(x_1, x_2, x_3, x_4, x_5)) = 2 + x_1 + 2*x_2 + 2*x_3 + 2*x_4 + x_5 11.84/3.91 POL(gt_in_gg(x_1, x_2)) = x_1 + x_2 11.84/3.91 POL(gt_out_gg(x_1, x_2)) = x_1 + x_2 11.84/3.91 POL(le_in_gg(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 11.84/3.91 POL(le_out_gg(x_1, x_2)) = 2*x_1 + 2*x_2 11.84/3.91 POL(s(x_1)) = 2*x_1 11.84/3.91 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (27) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) 11.84/3.91 U11_GGA(X, L1, Y, L2, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 11.84/3.91 The set Q consists of the following terms: 11.84/3.91 11.84/3.91 le_in_gg(x0, x1) 11.84/3.91 gt_in_gg(x0, x1) 11.84/3.91 U14_gg(x0, x1, x2) 11.84/3.91 U13_gg(x0, x1, x2) 11.84/3.91 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (28) DependencyGraphProof (EQUIVALENT) 11.84/3.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (29) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) 11.84/3.91 U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 11.84/3.91 The set Q consists of the following terms: 11.84/3.91 11.84/3.91 le_in_gg(x0, x1) 11.84/3.91 gt_in_gg(x0, x1) 11.84/3.91 U14_gg(x0, x1, x2) 11.84/3.91 U13_gg(x0, x1, x2) 11.84/3.91 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (30) UsableRulesProof (EQUIVALENT) 11.84/3.91 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. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (31) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) 11.84/3.91 U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 11.84/3.91 The set Q consists of the following terms: 11.84/3.91 11.84/3.91 le_in_gg(x0, x1) 11.84/3.91 gt_in_gg(x0, x1) 11.84/3.91 U14_gg(x0, x1, x2) 11.84/3.91 U13_gg(x0, x1, x2) 11.84/3.91 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (32) QReductionProof (EQUIVALENT) 11.84/3.91 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 11.84/3.91 11.84/3.91 gt_in_gg(x0, x1) 11.84/3.91 U13_gg(x0, x1, x2) 11.84/3.91 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (33) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) 11.84/3.91 U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 11.84/3.91 The set Q consists of the following terms: 11.84/3.91 11.84/3.91 le_in_gg(x0, x1) 11.84/3.91 U14_gg(x0, x1, x2) 11.84/3.91 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (34) QDPSizeChangeProof (EQUIVALENT) 11.84/3.91 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. 11.84/3.91 11.84/3.91 From the DPs we obtained the following set of size-change graphs: 11.84/3.91 *U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) 11.84/3.91 The graph contains the following edges 2 >= 1 11.84/3.91 11.84/3.91 11.84/3.91 *MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) 11.84/3.91 The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4 11.84/3.91 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (35) 11.84/3.91 YES 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (36) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> SPLIT_IN_GAA(L1, L2, L3) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> SPLIT2_IN_GAA(L1, L2, L3) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 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)) 11.84/3.91 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.91 11.84/3.91 [] = [] 11.84/3.91 11.84/3.91 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.91 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.91 11.84/3.91 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.91 11.84/3.91 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.91 11.84/3.91 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.91 11.84/3.91 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.91 11.84/3.91 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.91 11.84/3.91 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.91 11.84/3.91 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.91 11.84/3.91 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.91 11.84/3.91 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.91 11.84/3.91 s(x1) = s(x1) 11.84/3.91 11.84/3.91 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 0 = 0 11.84/3.91 11.84/3.91 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.91 11.84/3.91 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1) 11.84/3.91 11.84/3.91 SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (37) UsableRulesProof (EQUIVALENT) 11.84/3.91 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (38) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> SPLIT_IN_GAA(L1, L2, L3) 11.84/3.91 SPLIT_IN_GAA(L1, L2, L3) -> SPLIT2_IN_GAA(L1, L2, L3) 11.84/3.91 11.84/3.91 R is empty. 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1) 11.84/3.91 11.84/3.91 SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (39) PiDPToQDPProof (SOUND) 11.84/3.91 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (40) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 SPLIT2_IN_GAA(.(X, .(Y, L1))) -> SPLIT_IN_GAA(L1) 11.84/3.91 SPLIT_IN_GAA(L1) -> SPLIT2_IN_GAA(L1) 11.84/3.91 11.84/3.91 R is empty. 11.84/3.91 Q is empty. 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (41) QDPSizeChangeProof (EQUIVALENT) 11.84/3.91 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. 11.84/3.91 11.84/3.91 From the DPs we obtained the following set of size-change graphs: 11.84/3.91 *SPLIT_IN_GAA(L1) -> SPLIT2_IN_GAA(L1) 11.84/3.91 The graph contains the following edges 1 >= 1 11.84/3.91 11.84/3.91 11.84/3.91 *SPLIT2_IN_GAA(.(X, .(Y, L1))) -> SPLIT_IN_GAA(L1) 11.84/3.91 The graph contains the following edges 1 > 1 11.84/3.91 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (42) 11.84/3.91 YES 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (43) 11.84/3.91 Obligation: 11.84/3.91 Pi DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 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)) 11.84/3.91 U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4, L6) 11.84/3.91 MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3, L5) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([], []) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) 11.84/3.91 split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) 11.84/3.91 split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) 11.84/3.91 U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 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)) 11.84/3.91 U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) 11.84/3.91 U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) 11.84/3.91 merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 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)) 11.84/3.91 U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The argument filtering Pi contains the following mapping: 11.84/3.91 mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) 11.84/3.91 11.84/3.91 [] = [] 11.84/3.91 11.84/3.91 mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) 11.84/3.91 11.84/3.91 .(x1, x2) = .(x1, x2) 11.84/3.91 11.84/3.91 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) 11.84/3.91 11.84/3.91 U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) 11.84/3.91 11.84/3.91 split_in_gaa(x1, x2, x3) = split_in_gaa(x1) 11.84/3.91 11.84/3.91 U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) 11.84/3.91 11.84/3.91 split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) 11.84/3.91 11.84/3.91 split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) 11.84/3.91 11.84/3.91 split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) 11.84/3.91 11.84/3.91 split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) 11.84/3.91 11.84/3.91 split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) 11.84/3.91 11.84/3.91 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 11.84/3.91 11.84/3.91 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 11.84/3.91 11.84/3.91 U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 le_in_gg(x1, x2) = le_in_gg(x1, x2) 11.84/3.91 11.84/3.91 s(x1) = s(x1) 11.84/3.91 11.84/3.91 U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 0 = 0 11.84/3.91 11.84/3.91 le_out_gg(x1, x2) = le_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 11.84/3.91 11.84/3.91 U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) 11.84/3.91 11.84/3.91 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 11.84/3.91 11.84/3.91 U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 11.84/3.91 11.84/3.91 MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1) 11.84/3.91 11.84/3.91 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) 11.84/3.91 11.84/3.91 U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6) 11.84/3.91 11.84/3.91 11.84/3.91 We have to consider all (P,R,Pi)-chains 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (44) PiDPToQDPProof (SOUND) 11.84/3.91 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (45) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_GA(X, Y, L1, L4, mergesort_in_ga(L3)) 11.84/3.91 U2_GA(X, Y, L1, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4) 11.84/3.91 MERGESORT_IN_GA(.(X, .(Y, L1))) -> U1_GA(X, Y, L1, split2_in_gaa(.(X, .(Y, L1)))) 11.84/3.91 U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([]) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1))) -> U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1)))) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1))) -> U8_gaa(X, Y, L1, split_in_gaa(L1)) 11.84/3.91 split_in_gaa(L1) -> U5_gaa(L1, split0_in_gaa(L1)) 11.84/3.91 split0_in_gaa([]) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1) -> U6_gaa(L1, split1_in_gaa(L1)) 11.84/3.91 split1_in_gaa(.(X, [])) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1) -> U7_gaa(L1, split2_in_gaa(L1)) 11.84/3.91 U7_gaa(L1, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L4, mergesort_in_ga(L3)) 11.84/3.91 U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L5, mergesort_in_ga(L4)) 11.84/3.91 U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, merge_in_gga(L5, L6)) 11.84/3.91 merge_in_gga([], L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, []) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2)) -> U9_gga(X, L1, Y, L2, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 U9_gga(X, L1, Y, L2, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, merge_in_gga(L1, .(Y, L2))) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2)) -> U11_gga(X, L1, Y, L2, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 U11_gga(X, L1, Y, L2, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, merge_in_gga(.(X, L1), L2)) 11.84/3.91 U12_gga(X, L1, Y, L2, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The set Q consists of the following terms: 11.84/3.91 11.84/3.91 mergesort_in_ga(x0) 11.84/3.91 split2_in_gaa(x0) 11.84/3.91 split_in_gaa(x0) 11.84/3.91 split0_in_gaa(x0) 11.84/3.91 U5_gaa(x0, x1) 11.84/3.91 split1_in_gaa(x0) 11.84/3.91 U6_gaa(x0, x1) 11.84/3.91 U7_gaa(x0, x1) 11.84/3.91 U8_gaa(x0, x1, x2, x3) 11.84/3.91 U1_ga(x0, x1, x2, x3) 11.84/3.91 U2_ga(x0, x1, x2, x3, x4) 11.84/3.91 U3_ga(x0, x1, x2, x3, x4) 11.84/3.91 merge_in_gga(x0, x1) 11.84/3.91 le_in_gg(x0, x1) 11.84/3.91 U14_gg(x0, x1, x2) 11.84/3.91 U9_gga(x0, x1, x2, x3, x4) 11.84/3.91 gt_in_gg(x0, x1) 11.84/3.91 U13_gg(x0, x1, x2) 11.84/3.91 U11_gga(x0, x1, x2, x3, x4) 11.84/3.91 U12_gga(x0, x1, x2, x3, x4) 11.84/3.91 U10_gga(x0, x1, x2, x3, x4) 11.84/3.91 U4_ga(x0, x1, x2, x3) 11.84/3.91 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (46) QDPOrderProof (EQUIVALENT) 11.84/3.91 We use the reduction pair processor [LPAR04,JAR06]. 11.84/3.91 11.84/3.91 11.84/3.91 The following pairs can be oriented strictly and are deleted. 11.84/3.91 11.84/3.91 MERGESORT_IN_GA(.(X, .(Y, L1))) -> U1_GA(X, Y, L1, split2_in_gaa(.(X, .(Y, L1)))) 11.84/3.91 The remaining pairs can at least be oriented weakly. 11.84/3.91 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 11.84/3.91 11.84/3.91 POL( U2_GA_5(x_1, ..., x_5) ) = 2x_4 11.84/3.91 POL( mergesort_in_ga_1(x_1) ) = 0 11.84/3.91 POL( [] ) = 0 11.84/3.91 POL( mergesort_out_ga_2(x_1, x_2) ) = max{0, x_2 - 2} 11.84/3.91 POL( ._2(x_1, x_2) ) = 2x_2 + 1 11.84/3.91 POL( U1_ga_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2 11.84/3.91 POL( split2_in_gaa_1(x_1) ) = 2x_1 + 1 11.84/3.91 POL( U1_GA_4(x_1, ..., x_4) ) = max{0, x_4 - 2} 11.84/3.91 POL( U8_gaa_4(x_1, ..., x_4) ) = 2x_4 + 2 11.84/3.91 POL( split_in_gaa_1(x_1) ) = 2x_1 + 2 11.84/3.91 POL( split2_out_gaa_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 11.84/3.91 POL( U2_ga_5(x_1, ..., x_5) ) = max{0, x_1 + x_2 + x_4 - 2} 11.84/3.91 POL( U3_ga_5(x_1, ..., x_5) ) = 2x_4 + 2 11.84/3.91 POL( U4_ga_4(x_1, ..., x_4) ) = max{0, 2x_1 - 2} 11.84/3.91 POL( merge_in_gga_2(x_1, x_2) ) = max{0, x_1 - 2} 11.84/3.91 POL( U7_gaa_2(x_1, x_2) ) = x_2 11.84/3.91 POL( split_out_gaa_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 11.84/3.91 POL( U5_gaa_2(x_1, x_2) ) = x_2 + 2 11.84/3.91 POL( split0_in_gaa_1(x_1) ) = 0 11.84/3.91 POL( U6_gaa_2(x_1, x_2) ) = 2x_2 + 2 11.84/3.91 POL( split1_in_gaa_1(x_1) ) = x_1 11.84/3.91 POL( split0_out_gaa_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 11.84/3.91 POL( split1_out_gaa_3(x_1, ..., x_3) ) = x_2 + x_3 11.84/3.91 POL( merge_out_gga_3(x_1, ..., x_3) ) = max{0, x_2 - 2} 11.84/3.91 POL( U9_gga_5(x_1, ..., x_5) ) = x_2 + x_4 + 2 11.84/3.91 POL( le_in_gg_2(x_1, x_2) ) = 2x_1 11.84/3.91 POL( U11_gga_5(x_1, ..., x_5) ) = 2x_2 + 2x_3 + 2x_4 + 2 11.84/3.91 POL( gt_in_gg_2(x_1, x_2) ) = 0 11.84/3.91 POL( s_1(x_1) ) = 2x_1 11.84/3.91 POL( U14_gg_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2 11.84/3.91 POL( 0 ) = 0 11.84/3.91 POL( le_out_gg_2(x_1, x_2) ) = max{0, -2} 11.84/3.91 POL( U10_gga_5(x_1, ..., x_5) ) = 2x_1 + 2x_2 + 2 11.84/3.91 POL( U13_gg_3(x_1, ..., x_3) ) = 0 11.84/3.91 POL( gt_out_gg_2(x_1, x_2) ) = max{0, x_1 - 2} 11.84/3.91 POL( U12_gga_5(x_1, ..., x_5) ) = max{0, 2x_4 - 2} 11.84/3.91 POL( MERGESORT_IN_GA_1(x_1) ) = 2x_1 11.84/3.91 11.84/3.91 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 11.84/3.91 11.84/3.91 split2_in_gaa(.(X, .(Y, L1))) -> U8_gaa(X, Y, L1, split_in_gaa(L1)) 11.84/3.91 split_in_gaa(L1) -> U7_gaa(L1, split2_in_gaa(L1)) 11.84/3.91 U7_gaa(L1, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1) -> U5_gaa(L1, split0_in_gaa(L1)) 11.84/3.91 split_in_gaa(L1) -> U6_gaa(L1, split1_in_gaa(L1)) 11.84/3.91 U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 split0_in_gaa([]) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split1_in_gaa(.(X, [])) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 11.84/3.91 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (47) 11.84/3.91 Obligation: 11.84/3.91 Q DP problem: 11.84/3.91 The TRS P consists of the following rules: 11.84/3.91 11.84/3.91 U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_GA(X, Y, L1, L4, mergesort_in_ga(L3)) 11.84/3.91 U2_GA(X, Y, L1, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4) 11.84/3.91 U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3) 11.84/3.91 11.84/3.91 The TRS R consists of the following rules: 11.84/3.91 11.84/3.91 mergesort_in_ga([]) -> mergesort_out_ga([], []) 11.84/3.91 mergesort_in_ga(.(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) 11.84/3.91 mergesort_in_ga(.(X, .(Y, L1))) -> U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1)))) 11.84/3.91 split2_in_gaa(.(X, .(Y, L1))) -> U8_gaa(X, Y, L1, split_in_gaa(L1)) 11.84/3.91 split_in_gaa(L1) -> U5_gaa(L1, split0_in_gaa(L1)) 11.84/3.91 split0_in_gaa([]) -> split0_out_gaa([], [], []) 11.84/3.91 U5_gaa(L1, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1) -> U6_gaa(L1, split1_in_gaa(L1)) 11.84/3.91 split1_in_gaa(.(X, [])) -> split1_out_gaa(.(X, []), .(X, []), []) 11.84/3.91 U6_gaa(L1, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 split_in_gaa(L1) -> U7_gaa(L1, split2_in_gaa(L1)) 11.84/3.91 U7_gaa(L1, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) 11.84/3.91 U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) 11.84/3.91 U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L4, mergesort_in_ga(L3)) 11.84/3.91 U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L5, mergesort_in_ga(L4)) 11.84/3.91 U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, merge_in_gga(L5, L6)) 11.84/3.91 merge_in_gga([], L1) -> merge_out_gga([], L1, L1) 11.84/3.91 merge_in_gga(L1, []) -> merge_out_gga(L1, [], L1) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2)) -> U9_gga(X, L1, Y, L2, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) 11.84/3.91 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 11.84/3.91 le_in_gg(0, 0) -> le_out_gg(0, 0) 11.84/3.91 U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 11.84/3.91 U9_gga(X, L1, Y, L2, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, merge_in_gga(L1, .(Y, L2))) 11.84/3.91 merge_in_gga(.(X, L1), .(Y, L2)) -> U11_gga(X, L1, Y, L2, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) 11.84/3.91 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 11.84/3.91 U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 11.84/3.91 U11_gga(X, L1, Y, L2, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, merge_in_gga(.(X, L1), L2)) 11.84/3.91 U12_gga(X, L1, Y, L2, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) 11.84/3.91 U10_gga(X, L1, Y, L2, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) 11.84/3.91 U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) 11.84/3.91 11.84/3.91 The set Q consists of the following terms: 11.84/3.91 11.84/3.91 mergesort_in_ga(x0) 11.84/3.91 split2_in_gaa(x0) 11.84/3.91 split_in_gaa(x0) 11.84/3.91 split0_in_gaa(x0) 11.84/3.91 U5_gaa(x0, x1) 11.84/3.91 split1_in_gaa(x0) 11.84/3.91 U6_gaa(x0, x1) 11.84/3.91 U7_gaa(x0, x1) 11.84/3.91 U8_gaa(x0, x1, x2, x3) 11.84/3.91 U1_ga(x0, x1, x2, x3) 11.84/3.91 U2_ga(x0, x1, x2, x3, x4) 11.84/3.91 U3_ga(x0, x1, x2, x3, x4) 11.84/3.91 merge_in_gga(x0, x1) 11.84/3.91 le_in_gg(x0, x1) 11.84/3.91 U14_gg(x0, x1, x2) 11.84/3.91 U9_gga(x0, x1, x2, x3, x4) 11.84/3.91 gt_in_gg(x0, x1) 11.84/3.91 U13_gg(x0, x1, x2) 11.84/3.91 U11_gga(x0, x1, x2, x3, x4) 11.84/3.91 U12_gga(x0, x1, x2, x3, x4) 11.84/3.91 U10_gga(x0, x1, x2, x3, x4) 11.84/3.91 U4_ga(x0, x1, x2, x3) 11.84/3.91 11.84/3.91 We have to consider all (P,Q,R)-chains. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (48) DependencyGraphProof (EQUIVALENT) 11.84/3.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 11.84/3.91 ---------------------------------------- 11.84/3.91 11.84/3.91 (49) 11.84/3.91 TRUE 11.84/3.95 EOF