9.29/3.17 YES 9.29/3.19 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 9.29/3.19 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.29/3.19 9.29/3.19 9.29/3.19 Left Termination of the query pattern 9.29/3.19 9.29/3.19 qs(g,a) 9.29/3.19 9.29/3.19 w.r.t. the given Prolog program could successfully be proven: 9.29/3.19 9.29/3.19 (0) Prolog 9.29/3.19 (1) PrologToPiTRSProof [SOUND, 34 ms] 9.29/3.19 (2) PiTRS 9.29/3.19 (3) DependencyPairsProof [EQUIVALENT, 18 ms] 9.29/3.19 (4) PiDP 9.29/3.19 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 9.29/3.19 (6) AND 9.29/3.19 (7) PiDP 9.29/3.19 (8) UsableRulesProof [EQUIVALENT, 0 ms] 9.29/3.19 (9) PiDP 9.29/3.19 (10) PiDPToQDPProof [SOUND, 1 ms] 9.29/3.19 (11) QDP 9.29/3.19 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.29/3.19 (13) YES 9.29/3.19 (14) PiDP 9.29/3.19 (15) UsableRulesProof [EQUIVALENT, 0 ms] 9.29/3.19 (16) PiDP 9.29/3.19 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 9.29/3.19 (18) QDP 9.29/3.19 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.29/3.19 (20) YES 9.29/3.19 (21) PiDP 9.29/3.19 (22) UsableRulesProof [EQUIVALENT, 0 ms] 9.29/3.19 (23) PiDP 9.29/3.19 (24) PiDPToQDPProof [EQUIVALENT, 0 ms] 9.29/3.19 (25) QDP 9.29/3.19 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.29/3.19 (27) YES 9.29/3.19 (28) PiDP 9.29/3.19 (29) UsableRulesProof [EQUIVALENT, 0 ms] 9.29/3.19 (30) PiDP 9.29/3.19 (31) PiDPToQDPProof [SOUND, 0 ms] 9.29/3.19 (32) QDP 9.29/3.19 (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.29/3.19 (34) YES 9.29/3.19 (35) PiDP 9.29/3.19 (36) PiDPToQDPProof [SOUND, 0 ms] 9.29/3.19 (37) QDP 9.29/3.19 (38) QDPOrderProof [EQUIVALENT, 79 ms] 9.29/3.19 (39) QDP 9.29/3.19 (40) DependencyGraphProof [EQUIVALENT, 0 ms] 9.29/3.19 (41) TRUE 9.29/3.19 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (0) 9.29/3.19 Obligation: 9.29/3.19 Clauses: 9.29/3.19 9.29/3.19 qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))). 9.29/3.19 qs([], []). 9.29/3.19 part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(gt(X, Y), part(X, Xs, Ls, Bs)). 9.29/3.19 part(X, .(Y, Xs), Ls, .(Y, Bs)) :- ','(le(X, Y), part(X, Xs, Ls, Bs)). 9.29/3.19 part(X, [], [], []). 9.29/3.19 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 9.29/3.19 app([], Ys, Ys). 9.29/3.19 gt(s(X), s(Y)) :- gt(X, Y). 9.29/3.19 gt(s(0), 0). 9.29/3.19 le(s(X), s(Y)) :- le(X, Y). 9.29/3.19 le(0, s(0)). 9.29/3.19 le(0, 0). 9.29/3.19 9.29/3.19 9.29/3.19 Query: qs(g,a) 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (1) PrologToPiTRSProof (SOUND) 9.29/3.19 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 9.29/3.19 9.29/3.19 qs_in_2: (b,f) 9.29/3.19 9.29/3.19 part_in_4: (b,b,f,f) 9.29/3.19 9.29/3.19 gt_in_2: (b,b) 9.29/3.19 9.29/3.19 le_in_2: (b,b) 9.29/3.19 9.29/3.19 app_in_3: (b,b,f) 9.29/3.19 9.29/3.19 Transforming Prolog into the following Term Rewriting System: 9.29/3.19 9.29/3.19 Pi-finite rewrite system: 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 9.29/3.19 9.29/3.19 9.29/3.19 9.29/3.19 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 9.29/3.19 9.29/3.19 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (2) 9.29/3.19 Obligation: 9.29/3.19 Pi-finite rewrite system: 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (3) DependencyPairsProof (EQUIVALENT) 9.29/3.19 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GG(X, Y) 9.29/3.19 GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y)) 9.29/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.29/3.19 U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GG(X, Y) 9.29/3.19 LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y)) 9.29/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.29/3.19 U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 9.29/3.19 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) 9.29/3.19 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) 9.29/3.19 U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys) 9.29/3.19 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 QS_IN_GA(x1, x2) = QS_IN_GA(x1) 9.29/3.19 9.29/3.19 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 9.29/3.19 9.29/3.19 PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) 9.29/3.19 9.29/3.19 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 9.29/3.19 9.29/3.19 U10_GG(x1, x2, x3) = U10_GG(x3) 9.29/3.19 9.29/3.19 U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x2, x6) 9.29/3.19 9.29/3.19 U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 9.29/3.19 9.29/3.19 U11_GG(x1, x2, x3) = U11_GG(x3) 9.29/3.19 9.29/3.19 U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x2, x6) 9.29/3.19 9.29/3.19 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) 9.29/3.19 9.29/3.19 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_GA(x1, x2, x3, x4) = U4_GA(x4) 9.29/3.19 9.29/3.19 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 9.29/3.19 9.29/3.19 U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x5) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (4) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GG(X, Y) 9.29/3.19 GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y)) 9.29/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.29/3.19 U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GG(X, Y) 9.29/3.19 LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y)) 9.29/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.29/3.19 U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 9.29/3.19 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) 9.29/3.19 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) 9.29/3.19 U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys) 9.29/3.19 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 QS_IN_GA(x1, x2) = QS_IN_GA(x1) 9.29/3.19 9.29/3.19 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 9.29/3.19 9.29/3.19 PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) 9.29/3.19 9.29/3.19 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 9.29/3.19 9.29/3.19 U10_GG(x1, x2, x3) = U10_GG(x3) 9.29/3.19 9.29/3.19 U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x2, x6) 9.29/3.19 9.29/3.19 U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 9.29/3.19 9.29/3.19 U11_GG(x1, x2, x3) = U11_GG(x3) 9.29/3.19 9.29/3.19 U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x2, x6) 9.29/3.19 9.29/3.19 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) 9.29/3.19 9.29/3.19 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_GA(x1, x2, x3, x4) = U4_GA(x4) 9.29/3.19 9.29/3.19 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 9.29/3.19 9.29/3.19 U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x5) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (5) DependencyGraphProof (EQUIVALENT) 9.29/3.19 The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 11 less nodes. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (6) 9.29/3.19 Complex Obligation (AND) 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (7) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (8) UsableRulesProof (EQUIVALENT) 9.29/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (9) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 9.29/3.19 9.29/3.19 R is empty. 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (10) PiDPToQDPProof (SOUND) 9.29/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (11) 9.29/3.19 Obligation: 9.29/3.19 Q DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 9.29/3.19 9.29/3.19 R is empty. 9.29/3.19 Q is empty. 9.29/3.19 We have to consider all (P,Q,R)-chains. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (12) QDPSizeChangeProof (EQUIVALENT) 9.29/3.19 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. 9.29/3.19 9.29/3.19 From the DPs we obtained the following set of size-change graphs: 9.29/3.19 *APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 9.29/3.19 The graph contains the following edges 1 > 1, 2 >= 2 9.29/3.19 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (13) 9.29/3.19 YES 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (14) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (15) UsableRulesProof (EQUIVALENT) 9.29/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (16) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.29/3.19 9.29/3.19 R is empty. 9.29/3.19 Pi is empty. 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (17) PiDPToQDPProof (EQUIVALENT) 9.29/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (18) 9.29/3.19 Obligation: 9.29/3.19 Q DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.29/3.19 9.29/3.19 R is empty. 9.29/3.19 Q is empty. 9.29/3.19 We have to consider all (P,Q,R)-chains. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (19) QDPSizeChangeProof (EQUIVALENT) 9.29/3.19 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. 9.29/3.19 9.29/3.19 From the DPs we obtained the following set of size-change graphs: 9.29/3.19 *LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.29/3.19 The graph contains the following edges 1 > 1, 2 > 2 9.29/3.19 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (20) 9.29/3.19 YES 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (21) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (22) UsableRulesProof (EQUIVALENT) 9.29/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (23) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.29/3.19 9.29/3.19 R is empty. 9.29/3.19 Pi is empty. 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (24) PiDPToQDPProof (EQUIVALENT) 9.29/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (25) 9.29/3.19 Obligation: 9.29/3.19 Q DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.29/3.19 9.29/3.19 R is empty. 9.29/3.19 Q is empty. 9.29/3.19 We have to consider all (P,Q,R)-chains. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (26) QDPSizeChangeProof (EQUIVALENT) 9.29/3.19 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. 9.29/3.19 9.29/3.19 From the DPs we obtained the following set of size-change graphs: 9.29/3.19 *GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.29/3.19 The graph contains the following edges 1 > 1, 2 > 2 9.29/3.19 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (27) 9.29/3.19 YES 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (28) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) 9.29/3.19 9.29/3.19 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (29) UsableRulesProof (EQUIVALENT) 9.29/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (30) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) 9.29/3.19 9.29/3.19 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (31) PiDPToQDPProof (SOUND) 9.29/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (32) 9.29/3.19 Obligation: 9.29/3.19 Q DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 U5_GGAA(X, Y, Xs, gt_out_gg) -> PART_IN_GGAA(X, Xs) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, gt_in_gg(X, Y)) 9.29/3.19 PART_IN_GGAA(X, .(Y, Xs)) -> U7_GGAA(X, Y, Xs, le_in_gg(X, Y)) 9.29/3.19 U7_GGAA(X, Y, Xs, le_out_gg) -> PART_IN_GGAA(X, Xs) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg 9.29/3.19 le_in_gg(0, 0) -> le_out_gg 9.29/3.19 U10_gg(gt_out_gg) -> gt_out_gg 9.29/3.19 U11_gg(le_out_gg) -> le_out_gg 9.29/3.19 9.29/3.19 The set Q consists of the following terms: 9.29/3.19 9.29/3.19 gt_in_gg(x0, x1) 9.29/3.19 le_in_gg(x0, x1) 9.29/3.19 U10_gg(x0) 9.29/3.19 U11_gg(x0) 9.29/3.19 9.29/3.19 We have to consider all (P,Q,R)-chains. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (33) QDPSizeChangeProof (EQUIVALENT) 9.29/3.19 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. 9.29/3.19 9.29/3.19 From the DPs we obtained the following set of size-change graphs: 9.29/3.19 *PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, gt_in_gg(X, Y)) 9.29/3.19 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 9.29/3.19 9.29/3.19 9.29/3.19 *PART_IN_GGAA(X, .(Y, Xs)) -> U7_GGAA(X, Y, Xs, le_in_gg(X, Y)) 9.29/3.19 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 9.29/3.19 9.29/3.19 9.29/3.19 *U5_GGAA(X, Y, Xs, gt_out_gg) -> PART_IN_GGAA(X, Xs) 9.29/3.19 The graph contains the following edges 1 >= 1, 3 >= 2 9.29/3.19 9.29/3.19 9.29/3.19 *U7_GGAA(X, Y, Xs, le_out_gg) -> PART_IN_GGAA(X, Xs) 9.29/3.19 The graph contains the following edges 1 >= 1, 3 >= 2 9.29/3.19 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (34) 9.29/3.19 YES 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (35) 9.29/3.19 Obligation: 9.29/3.19 Pi DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) 9.29/3.19 QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) 9.29/3.19 U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.29/3.19 U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) 9.29/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.29/3.19 U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.29/3.19 U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 9.29/3.19 part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) 9.29/3.19 U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 9.29/3.19 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 9.29/3.19 qs_in_ga([], []) -> qs_out_ga([], []) 9.29/3.19 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 9.29/3.19 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 9.29/3.19 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 9.29/3.19 app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) 9.29/3.19 U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 9.29/3.19 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 9.29/3.19 9.29/3.19 The argument filtering Pi contains the following mapping: 9.29/3.19 qs_in_ga(x1, x2) = qs_in_ga(x1) 9.29/3.19 9.29/3.19 .(x1, x2) = .(x1, x2) 9.29/3.19 9.29/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.29/3.19 9.29/3.19 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 9.29/3.19 9.29/3.19 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.29/3.19 9.29/3.19 s(x1) = s(x1) 9.29/3.19 9.29/3.19 U10_gg(x1, x2, x3) = U10_gg(x3) 9.29/3.19 9.29/3.19 0 = 0 9.29/3.19 9.29/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.29/3.19 9.29/3.19 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 9.29/3.19 9.29/3.19 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) 9.29/3.19 9.29/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.29/3.19 9.29/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.29/3.19 9.29/3.19 le_out_gg(x1, x2) = le_out_gg 9.29/3.19 9.29/3.19 U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) 9.29/3.19 9.29/3.19 [] = [] 9.29/3.19 9.29/3.19 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 9.29/3.19 9.29/3.19 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 qs_out_ga(x1, x2) = qs_out_ga(x2) 9.29/3.19 9.29/3.19 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 9.29/3.19 9.29/3.19 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 9.29/3.19 9.29/3.19 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 9.29/3.19 9.29/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) 9.29/3.19 9.29/3.19 app_out_gga(x1, x2, x3) = app_out_gga(x3) 9.29/3.19 9.29/3.19 QS_IN_GA(x1, x2) = QS_IN_GA(x1) 9.29/3.19 9.29/3.19 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 9.29/3.19 9.29/3.19 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) 9.29/3.19 9.29/3.19 9.29/3.19 We have to consider all (P,R,Pi)-chains 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (36) PiDPToQDPProof (SOUND) 9.29/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (37) 9.29/3.19 Obligation: 9.29/3.19 Q DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 U1_GA(X, part_out_ggaa(Littles, Bigs)) -> U2_GA(X, Bigs, qs_in_ga(Littles)) 9.29/3.19 U2_GA(X, Bigs, qs_out_ga(Ls)) -> QS_IN_GA(Bigs) 9.29/3.19 QS_IN_GA(.(X, Xs)) -> U1_GA(X, part_in_ggaa(X, Xs)) 9.29/3.19 U1_GA(X, part_out_ggaa(Littles, Bigs)) -> QS_IN_GA(Littles) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs)) -> U1_ga(X, part_in_ggaa(X, Xs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg 9.29/3.19 U10_gg(gt_out_gg) -> gt_out_gg 9.29/3.19 U5_ggaa(X, Y, Xs, gt_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg 9.29/3.19 le_in_gg(0, 0) -> le_out_gg 9.29/3.19 U11_gg(le_out_gg) -> le_out_gg 9.29/3.19 U7_ggaa(X, Y, Xs, le_out_gg) -> U8_ggaa(Y, part_in_ggaa(X, Xs)) 9.29/3.19 part_in_ggaa(X, []) -> part_out_ggaa([], []) 9.29/3.19 U8_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) 9.29/3.19 U1_ga(X, part_out_ggaa(Littles, Bigs)) -> U2_ga(X, Bigs, qs_in_ga(Littles)) 9.29/3.19 qs_in_ga([]) -> qs_out_ga([]) 9.29/3.19 U2_ga(X, Bigs, qs_out_ga(Ls)) -> U3_ga(X, Ls, qs_in_ga(Bigs)) 9.29/3.19 U3_ga(X, Ls, qs_out_ga(Bs)) -> U4_ga(app_in_gga(Ls, .(X, Bs))) 9.29/3.19 app_in_gga(.(X, Xs), Ys) -> U9_gga(X, app_in_gga(Xs, Ys)) 9.29/3.19 app_in_gga([], Ys) -> app_out_gga(Ys) 9.29/3.19 U9_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) 9.29/3.19 U4_ga(app_out_gga(Ys)) -> qs_out_ga(Ys) 9.29/3.19 9.29/3.19 The set Q consists of the following terms: 9.29/3.19 9.29/3.19 qs_in_ga(x0) 9.29/3.19 part_in_ggaa(x0, x1) 9.29/3.19 gt_in_gg(x0, x1) 9.29/3.19 U10_gg(x0) 9.29/3.19 U5_ggaa(x0, x1, x2, x3) 9.29/3.19 le_in_gg(x0, x1) 9.29/3.19 U11_gg(x0) 9.29/3.19 U7_ggaa(x0, x1, x2, x3) 9.29/3.19 U8_ggaa(x0, x1) 9.29/3.19 U6_ggaa(x0, x1) 9.29/3.19 U1_ga(x0, x1) 9.29/3.19 U2_ga(x0, x1, x2) 9.29/3.19 U3_ga(x0, x1, x2) 9.29/3.19 app_in_gga(x0, x1) 9.29/3.19 U9_gga(x0, x1) 9.29/3.19 U4_ga(x0) 9.29/3.19 9.29/3.19 We have to consider all (P,Q,R)-chains. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (38) QDPOrderProof (EQUIVALENT) 9.29/3.19 We use the reduction pair processor [LPAR04,JAR06]. 9.29/3.19 9.29/3.19 9.29/3.19 The following pairs can be oriented strictly and are deleted. 9.29/3.19 9.29/3.19 U1_GA(X, part_out_ggaa(Littles, Bigs)) -> U2_GA(X, Bigs, qs_in_ga(Littles)) 9.29/3.19 U1_GA(X, part_out_ggaa(Littles, Bigs)) -> QS_IN_GA(Littles) 9.29/3.19 The remaining pairs can at least be oriented weakly. 9.29/3.19 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 9.29/3.19 9.29/3.19 POL( U2_GA_3(x_1, ..., x_3) ) = x_2 + 1 9.29/3.19 POL( qs_in_ga_1(x_1) ) = 2 9.29/3.19 POL( ._2(x_1, x_2) ) = x_2 + 1 9.29/3.19 POL( U1_ga_2(x_1, x_2) ) = 2x_1 + 2 9.29/3.19 POL( part_in_ggaa_2(x_1, x_2) ) = x_2 9.29/3.19 POL( [] ) = 0 9.29/3.19 POL( qs_out_ga_1(x_1) ) = max{0, x_1 - 2} 9.29/3.19 POL( U1_GA_2(x_1, x_2) ) = x_2 + 2 9.29/3.19 POL( U5_ggaa_4(x_1, ..., x_4) ) = x_3 + 1 9.29/3.19 POL( gt_in_gg_2(x_1, x_2) ) = 0 9.29/3.19 POL( U7_ggaa_4(x_1, ..., x_4) ) = x_3 + 1 9.29/3.19 POL( le_in_gg_2(x_1, x_2) ) = 0 9.29/3.19 POL( part_out_ggaa_2(x_1, x_2) ) = x_1 + x_2 9.29/3.19 POL( U2_ga_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2 9.29/3.19 POL( U3_ga_3(x_1, ..., x_3) ) = max{0, -2} 9.29/3.19 POL( U4_ga_1(x_1) ) = 2 9.29/3.19 POL( app_in_gga_2(x_1, x_2) ) = max{0, 2x_2 - 2} 9.29/3.19 POL( gt_out_gg ) = 0 9.29/3.19 POL( U6_ggaa_2(x_1, x_2) ) = x_2 + 1 9.29/3.19 POL( s_1(x_1) ) = 0 9.29/3.19 POL( U10_gg_1(x_1) ) = 2 9.29/3.19 POL( 0 ) = 0 9.29/3.19 POL( U11_gg_1(x_1) ) = 2 9.29/3.19 POL( le_out_gg ) = 0 9.29/3.19 POL( U8_ggaa_2(x_1, x_2) ) = x_2 + 1 9.29/3.19 POL( U9_gga_2(x_1, x_2) ) = x_1 + 2 9.29/3.19 POL( app_out_gga_1(x_1) ) = max{0, x_1 - 2} 9.29/3.19 POL( QS_IN_GA_1(x_1) ) = x_1 + 1 9.29/3.19 9.29/3.19 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 9.29/3.19 9.29/3.19 part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y)) 9.29/3.19 part_in_ggaa(X, []) -> part_out_ggaa([], []) 9.29/3.19 U5_ggaa(X, Y, Xs, gt_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) 9.29/3.19 U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) 9.29/3.19 U7_ggaa(X, Y, Xs, le_out_gg) -> U8_ggaa(Y, part_in_ggaa(X, Xs)) 9.29/3.19 U8_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) 9.29/3.19 9.29/3.19 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (39) 9.29/3.19 Obligation: 9.29/3.19 Q DP problem: 9.29/3.19 The TRS P consists of the following rules: 9.29/3.19 9.29/3.19 U2_GA(X, Bigs, qs_out_ga(Ls)) -> QS_IN_GA(Bigs) 9.29/3.19 QS_IN_GA(.(X, Xs)) -> U1_GA(X, part_in_ggaa(X, Xs)) 9.29/3.19 9.29/3.19 The TRS R consists of the following rules: 9.29/3.19 9.29/3.19 qs_in_ga(.(X, Xs)) -> U1_ga(X, part_in_ggaa(X, Xs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) 9.29/3.19 gt_in_gg(s(0), 0) -> gt_out_gg 9.29/3.19 U10_gg(gt_out_gg) -> gt_out_gg 9.29/3.19 U5_ggaa(X, Y, Xs, gt_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) 9.29/3.19 part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y)) 9.29/3.19 le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) 9.29/3.19 le_in_gg(0, s(0)) -> le_out_gg 9.29/3.19 le_in_gg(0, 0) -> le_out_gg 9.29/3.19 U11_gg(le_out_gg) -> le_out_gg 9.29/3.19 U7_ggaa(X, Y, Xs, le_out_gg) -> U8_ggaa(Y, part_in_ggaa(X, Xs)) 9.29/3.19 part_in_ggaa(X, []) -> part_out_ggaa([], []) 9.29/3.19 U8_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) 9.29/3.19 U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) 9.29/3.19 U1_ga(X, part_out_ggaa(Littles, Bigs)) -> U2_ga(X, Bigs, qs_in_ga(Littles)) 9.29/3.19 qs_in_ga([]) -> qs_out_ga([]) 9.29/3.19 U2_ga(X, Bigs, qs_out_ga(Ls)) -> U3_ga(X, Ls, qs_in_ga(Bigs)) 9.29/3.19 U3_ga(X, Ls, qs_out_ga(Bs)) -> U4_ga(app_in_gga(Ls, .(X, Bs))) 9.29/3.19 app_in_gga(.(X, Xs), Ys) -> U9_gga(X, app_in_gga(Xs, Ys)) 9.29/3.19 app_in_gga([], Ys) -> app_out_gga(Ys) 9.29/3.19 U9_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) 9.29/3.19 U4_ga(app_out_gga(Ys)) -> qs_out_ga(Ys) 9.29/3.19 9.29/3.19 The set Q consists of the following terms: 9.29/3.19 9.29/3.19 qs_in_ga(x0) 9.29/3.19 part_in_ggaa(x0, x1) 9.29/3.19 gt_in_gg(x0, x1) 9.29/3.19 U10_gg(x0) 9.29/3.19 U5_ggaa(x0, x1, x2, x3) 9.29/3.19 le_in_gg(x0, x1) 9.29/3.19 U11_gg(x0) 9.29/3.19 U7_ggaa(x0, x1, x2, x3) 9.29/3.19 U8_ggaa(x0, x1) 9.29/3.19 U6_ggaa(x0, x1) 9.29/3.19 U1_ga(x0, x1) 9.29/3.19 U2_ga(x0, x1, x2) 9.29/3.19 U3_ga(x0, x1, x2) 9.29/3.19 app_in_gga(x0, x1) 9.29/3.19 U9_gga(x0, x1) 9.29/3.19 U4_ga(x0) 9.29/3.19 9.29/3.19 We have to consider all (P,Q,R)-chains. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (40) DependencyGraphProof (EQUIVALENT) 9.29/3.19 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 9.29/3.19 ---------------------------------------- 9.29/3.19 9.29/3.19 (41) 9.29/3.19 TRUE 9.44/3.25 EOF