7.70/2.85 YES 7.70/2.89 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 7.70/2.89 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.70/2.89 7.70/2.89 7.70/2.89 Left Termination of the query pattern 7.70/2.89 7.70/2.89 qs(g,a) 7.70/2.89 7.70/2.89 w.r.t. the given Prolog program could successfully be proven: 7.70/2.89 7.70/2.89 (0) Prolog 7.70/2.89 (1) PrologToPiTRSProof [SOUND, 0 ms] 7.70/2.89 (2) PiTRS 7.70/2.89 (3) DependencyPairsProof [EQUIVALENT, 2 ms] 7.70/2.89 (4) PiDP 7.70/2.89 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 7.70/2.89 (6) AND 7.70/2.89 (7) PiDP 7.70/2.89 (8) UsableRulesProof [EQUIVALENT, 0 ms] 7.70/2.89 (9) PiDP 7.70/2.89 (10) PiDPToQDPProof [SOUND, 14 ms] 7.70/2.89 (11) QDP 7.70/2.89 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.70/2.89 (13) YES 7.70/2.89 (14) PiDP 7.70/2.89 (15) UsableRulesProof [EQUIVALENT, 0 ms] 7.70/2.89 (16) PiDP 7.70/2.89 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 7.70/2.89 (18) QDP 7.70/2.89 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.70/2.89 (20) YES 7.70/2.89 (21) PiDP 7.70/2.89 (22) UsableRulesProof [EQUIVALENT, 0 ms] 7.70/2.89 (23) PiDP 7.70/2.89 (24) PiDPToQDPProof [SOUND, 0 ms] 7.70/2.89 (25) QDP 7.70/2.89 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.70/2.89 (27) YES 7.70/2.89 (28) PiDP 7.70/2.89 (29) PiDPToQDPProof [SOUND, 0 ms] 7.70/2.89 (30) QDP 7.70/2.89 (31) QDPOrderProof [EQUIVALENT, 72 ms] 7.70/2.89 (32) QDP 7.70/2.89 (33) DependencyGraphProof [EQUIVALENT, 1 ms] 7.70/2.89 (34) TRUE 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (0) 7.70/2.89 Obligation: 7.70/2.89 Clauses: 7.70/2.89 7.70/2.89 qs([], []). 7.70/2.89 qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))). 7.70/2.89 part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)). 7.70/2.89 part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs). 7.70/2.89 part(X1, [], [], []). 7.70/2.89 app([], X, X). 7.70/2.89 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 7.70/2.89 less(0, s(X2)). 7.70/2.89 less(s(X), s(Y)) :- less(X, Y). 7.70/2.89 7.70/2.89 7.70/2.89 Query: qs(g,a) 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (1) PrologToPiTRSProof (SOUND) 7.70/2.89 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 7.70/2.89 7.70/2.89 qs_in_2: (b,f) 7.70/2.89 7.70/2.89 part_in_4: (b,b,f,f) 7.70/2.89 7.70/2.89 less_in_2: (b,b) 7.70/2.89 7.70/2.89 app_in_3: (b,b,f) 7.70/2.89 7.70/2.89 Transforming Prolog into the following Term Rewriting System: 7.70/2.89 7.70/2.89 Pi-finite rewrite system: 7.70/2.89 The TRS R consists of the following rules: 7.70/2.89 7.70/2.89 qs_in_ga([], []) -> qs_out_ga([], []) 7.70/2.89 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.89 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.89 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.89 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.89 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.89 U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 part_in_ggaa(X1, [], [], []) -> part_out_ggaa(X1, [], [], []) 7.70/2.89 U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 7.70/2.89 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 7.70/2.89 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.89 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.89 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.89 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.70/2.89 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.89 U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.70/2.89 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 7.70/2.89 7.70/2.89 The argument filtering Pi contains the following mapping: 7.70/2.89 qs_in_ga(x1, x2) = qs_in_ga(x1) 7.70/2.89 7.70/2.89 [] = [] 7.70/2.89 7.70/2.89 qs_out_ga(x1, x2) = qs_out_ga(x2) 7.70/2.89 7.70/2.89 .(x1, x2) = .(x1, x2) 7.70/2.89 7.70/2.89 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 7.70/2.89 7.70/2.89 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 7.70/2.89 7.70/2.89 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 7.70/2.89 7.70/2.89 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.89 7.70/2.89 0 = 0 7.70/2.89 7.70/2.89 s(x1) = s(x1) 7.70/2.89 7.70/2.89 less_out_gg(x1, x2) = less_out_gg 7.70/2.89 7.70/2.89 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.89 7.70/2.89 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 7.70/2.89 7.70/2.89 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6) 7.70/2.89 7.70/2.89 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 7.70/2.89 7.70/2.89 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 7.70/2.89 7.70/2.89 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 7.70/2.89 7.70/2.89 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 7.70/2.89 7.70/2.89 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.70/2.89 7.70/2.89 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.70/2.89 7.70/2.89 U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5) 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (2) 7.70/2.89 Obligation: 7.70/2.89 Pi-finite rewrite system: 7.70/2.89 The TRS R consists of the following rules: 7.70/2.89 7.70/2.89 qs_in_ga([], []) -> qs_out_ga([], []) 7.70/2.89 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.89 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.89 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.89 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.89 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.89 U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 part_in_ggaa(X1, [], [], []) -> part_out_ggaa(X1, [], [], []) 7.70/2.89 U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 7.70/2.89 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 7.70/2.89 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.89 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.89 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.89 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.70/2.89 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.89 U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.70/2.89 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 7.70/2.89 7.70/2.89 The argument filtering Pi contains the following mapping: 7.70/2.89 qs_in_ga(x1, x2) = qs_in_ga(x1) 7.70/2.89 7.70/2.89 [] = [] 7.70/2.89 7.70/2.89 qs_out_ga(x1, x2) = qs_out_ga(x2) 7.70/2.89 7.70/2.89 .(x1, x2) = .(x1, x2) 7.70/2.89 7.70/2.89 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 7.70/2.89 7.70/2.89 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 7.70/2.89 7.70/2.89 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 7.70/2.89 7.70/2.89 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.89 7.70/2.89 0 = 0 7.70/2.89 7.70/2.89 s(x1) = s(x1) 7.70/2.89 7.70/2.89 less_out_gg(x1, x2) = less_out_gg 7.70/2.89 7.70/2.89 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.89 7.70/2.89 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 7.70/2.89 7.70/2.89 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6) 7.70/2.89 7.70/2.89 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 7.70/2.89 7.70/2.89 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 7.70/2.89 7.70/2.89 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 7.70/2.89 7.70/2.89 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 7.70/2.89 7.70/2.89 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.70/2.89 7.70/2.89 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.70/2.89 7.70/2.89 U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5) 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (3) DependencyPairsProof (EQUIVALENT) 7.70/2.89 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 7.70/2.89 Pi DP problem: 7.70/2.89 The TRS P consists of the following rules: 7.70/2.89 7.70/2.89 QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.89 QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs) 7.70/2.89 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.89 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GG(X, Y) 7.70/2.89 LESS_IN_GG(s(X), s(Y)) -> U9_GG(X, Y, less_in_gg(X, Y)) 7.70/2.89 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 7.70/2.89 U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 7.70/2.89 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 7.70/2.89 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.89 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) 7.70/2.89 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.89 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) 7.70/2.89 U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.89 U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys) 7.70/2.89 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U8_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.89 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.70/2.89 7.70/2.89 The TRS R consists of the following rules: 7.70/2.89 7.70/2.89 qs_in_ga([], []) -> qs_out_ga([], []) 7.70/2.89 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.89 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.89 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.89 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.89 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.89 U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 part_in_ggaa(X1, [], [], []) -> part_out_ggaa(X1, [], [], []) 7.70/2.89 U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 7.70/2.89 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 7.70/2.89 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.89 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.89 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.89 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.70/2.89 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.89 U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.70/2.89 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 7.70/2.89 7.70/2.89 The argument filtering Pi contains the following mapping: 7.70/2.89 qs_in_ga(x1, x2) = qs_in_ga(x1) 7.70/2.89 7.70/2.89 [] = [] 7.70/2.89 7.70/2.89 qs_out_ga(x1, x2) = qs_out_ga(x2) 7.70/2.89 7.70/2.89 .(x1, x2) = .(x1, x2) 7.70/2.89 7.70/2.89 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 7.70/2.89 7.70/2.89 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 7.70/2.89 7.70/2.89 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 7.70/2.89 7.70/2.89 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.89 7.70/2.89 0 = 0 7.70/2.89 7.70/2.89 s(x1) = s(x1) 7.70/2.89 7.70/2.89 less_out_gg(x1, x2) = less_out_gg 7.70/2.89 7.70/2.89 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.89 7.70/2.89 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 7.70/2.89 7.70/2.89 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6) 7.70/2.89 7.70/2.89 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 7.70/2.89 7.70/2.89 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 7.70/2.89 7.70/2.89 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 7.70/2.89 7.70/2.89 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 7.70/2.89 7.70/2.89 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.70/2.89 7.70/2.89 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.70/2.89 7.70/2.89 U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5) 7.70/2.89 7.70/2.89 QS_IN_GA(x1, x2) = QS_IN_GA(x1) 7.70/2.89 7.70/2.89 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 7.70/2.89 7.70/2.89 PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) 7.70/2.89 7.70/2.89 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 7.70/2.89 7.70/2.89 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 7.70/2.89 7.70/2.89 U9_GG(x1, x2, x3) = U9_GG(x3) 7.70/2.89 7.70/2.89 U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x2, x6) 7.70/2.89 7.70/2.89 U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x2, x6) 7.70/2.89 7.70/2.89 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) 7.70/2.89 7.70/2.89 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5) 7.70/2.89 7.70/2.89 U4_GA(x1, x2, x3, x4) = U4_GA(x4) 7.70/2.89 7.70/2.89 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.70/2.89 7.70/2.89 U8_GGA(x1, x2, x3, x4, x5) = U8_GGA(x1, x5) 7.70/2.89 7.70/2.89 7.70/2.89 We have to consider all (P,R,Pi)-chains 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (4) 7.70/2.89 Obligation: 7.70/2.89 Pi DP problem: 7.70/2.89 The TRS P consists of the following rules: 7.70/2.89 7.70/2.89 QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.89 QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs) 7.70/2.89 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.89 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GG(X, Y) 7.70/2.89 LESS_IN_GG(s(X), s(Y)) -> U9_GG(X, Y, less_in_gg(X, Y)) 7.70/2.89 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 7.70/2.89 U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 7.70/2.89 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 7.70/2.89 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.89 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) 7.70/2.89 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.89 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) 7.70/2.89 U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.89 U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys) 7.70/2.89 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U8_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.89 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.70/2.89 7.70/2.89 The TRS R consists of the following rules: 7.70/2.89 7.70/2.89 qs_in_ga([], []) -> qs_out_ga([], []) 7.70/2.89 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.89 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.89 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.89 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.89 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.89 U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.89 part_in_ggaa(X1, [], [], []) -> part_out_ggaa(X1, [], [], []) 7.70/2.89 U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 7.70/2.89 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 7.70/2.89 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.89 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.89 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.89 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.70/2.89 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.89 U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.70/2.89 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 7.70/2.89 7.70/2.89 The argument filtering Pi contains the following mapping: 7.70/2.89 qs_in_ga(x1, x2) = qs_in_ga(x1) 7.70/2.89 7.70/2.89 [] = [] 7.70/2.89 7.70/2.89 qs_out_ga(x1, x2) = qs_out_ga(x2) 7.70/2.89 7.70/2.89 .(x1, x2) = .(x1, x2) 7.70/2.89 7.70/2.89 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 7.70/2.89 7.70/2.89 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 7.70/2.89 7.70/2.89 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 7.70/2.89 7.70/2.89 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.89 7.70/2.89 0 = 0 7.70/2.89 7.70/2.89 s(x1) = s(x1) 7.70/2.89 7.70/2.89 less_out_gg(x1, x2) = less_out_gg 7.70/2.89 7.70/2.89 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.89 7.70/2.89 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 7.70/2.89 7.70/2.89 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6) 7.70/2.89 7.70/2.89 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 7.70/2.89 7.70/2.89 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 7.70/2.89 7.70/2.89 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 7.70/2.89 7.70/2.89 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 7.70/2.89 7.70/2.89 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.70/2.89 7.70/2.89 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.70/2.89 7.70/2.89 U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5) 7.70/2.89 7.70/2.89 QS_IN_GA(x1, x2) = QS_IN_GA(x1) 7.70/2.89 7.70/2.89 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 7.70/2.89 7.70/2.89 PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) 7.70/2.89 7.70/2.89 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 7.70/2.89 7.70/2.89 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 7.70/2.89 7.70/2.89 U9_GG(x1, x2, x3) = U9_GG(x3) 7.70/2.89 7.70/2.89 U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x2, x6) 7.70/2.89 7.70/2.89 U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x2, x6) 7.70/2.89 7.70/2.89 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) 7.70/2.89 7.70/2.89 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5) 7.70/2.89 7.70/2.89 U4_GA(x1, x2, x3, x4) = U4_GA(x4) 7.70/2.89 7.70/2.89 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.70/2.89 7.70/2.89 U8_GGA(x1, x2, x3, x4, x5) = U8_GGA(x1, x5) 7.70/2.89 7.70/2.89 7.70/2.89 We have to consider all (P,R,Pi)-chains 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (5) DependencyGraphProof (EQUIVALENT) 7.70/2.90 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (6) 7.70/2.90 Complex Obligation (AND) 7.70/2.90 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (7) 7.70/2.90 Obligation: 7.70/2.90 Pi DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.70/2.90 7.70/2.90 The TRS R consists of the following rules: 7.70/2.90 7.70/2.90 qs_in_ga([], []) -> qs_out_ga([], []) 7.70/2.90 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.90 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.90 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.90 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.90 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.90 U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.90 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.90 part_in_ggaa(X1, [], [], []) -> part_out_ggaa(X1, [], [], []) 7.70/2.90 U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 7.70/2.90 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 7.70/2.90 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.90 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.90 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.90 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.70/2.90 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.90 U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.70/2.90 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 7.70/2.90 7.70/2.90 The argument filtering Pi contains the following mapping: 7.70/2.90 qs_in_ga(x1, x2) = qs_in_ga(x1) 7.70/2.90 7.70/2.90 [] = [] 7.70/2.90 7.70/2.90 qs_out_ga(x1, x2) = qs_out_ga(x2) 7.70/2.90 7.70/2.90 .(x1, x2) = .(x1, x2) 7.70/2.90 7.70/2.90 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 7.70/2.90 7.70/2.90 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 7.70/2.90 7.70/2.90 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 7.70/2.90 7.70/2.90 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.90 7.70/2.90 0 = 0 7.70/2.90 7.70/2.90 s(x1) = s(x1) 7.70/2.90 7.70/2.90 less_out_gg(x1, x2) = less_out_gg 7.70/2.90 7.70/2.90 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.90 7.70/2.90 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 7.70/2.90 7.70/2.90 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6) 7.70/2.90 7.70/2.90 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 7.70/2.90 7.70/2.90 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 7.70/2.90 7.70/2.90 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 7.70/2.90 7.70/2.90 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 7.70/2.90 7.70/2.90 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.70/2.90 7.70/2.90 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.70/2.90 7.70/2.90 U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5) 7.70/2.90 7.70/2.90 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.70/2.90 7.70/2.90 7.70/2.90 We have to consider all (P,R,Pi)-chains 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (8) UsableRulesProof (EQUIVALENT) 7.70/2.90 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (9) 7.70/2.90 Obligation: 7.70/2.90 Pi DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 7.70/2.90 7.70/2.90 R is empty. 7.70/2.90 The argument filtering Pi contains the following mapping: 7.70/2.90 .(x1, x2) = .(x1, x2) 7.70/2.90 7.70/2.90 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 7.70/2.90 7.70/2.90 7.70/2.90 We have to consider all (P,R,Pi)-chains 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (10) PiDPToQDPProof (SOUND) 7.70/2.90 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (11) 7.70/2.90 Obligation: 7.70/2.90 Q DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 7.70/2.90 7.70/2.90 R is empty. 7.70/2.90 Q is empty. 7.70/2.90 We have to consider all (P,Q,R)-chains. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (12) QDPSizeChangeProof (EQUIVALENT) 7.70/2.90 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. 7.70/2.90 7.70/2.90 From the DPs we obtained the following set of size-change graphs: 7.70/2.90 *APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 7.70/2.90 The graph contains the following edges 1 > 1, 2 >= 2 7.70/2.90 7.70/2.90 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (13) 7.70/2.90 YES 7.70/2.90 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (14) 7.70/2.90 Obligation: 7.70/2.90 Pi DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 7.70/2.90 7.70/2.90 The TRS R consists of the following rules: 7.70/2.90 7.70/2.90 qs_in_ga([], []) -> qs_out_ga([], []) 7.70/2.90 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.90 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.90 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.90 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.90 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.90 U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.90 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.90 part_in_ggaa(X1, [], [], []) -> part_out_ggaa(X1, [], [], []) 7.70/2.90 U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 7.70/2.90 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 7.70/2.90 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.90 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.90 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.90 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.70/2.90 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.90 U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.70/2.90 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 7.70/2.90 7.70/2.90 The argument filtering Pi contains the following mapping: 7.70/2.90 qs_in_ga(x1, x2) = qs_in_ga(x1) 7.70/2.90 7.70/2.90 [] = [] 7.70/2.90 7.70/2.90 qs_out_ga(x1, x2) = qs_out_ga(x2) 7.70/2.90 7.70/2.90 .(x1, x2) = .(x1, x2) 7.70/2.90 7.70/2.90 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 7.70/2.90 7.70/2.90 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 7.70/2.90 7.70/2.90 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 7.70/2.90 7.70/2.90 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.90 7.70/2.90 0 = 0 7.70/2.90 7.70/2.90 s(x1) = s(x1) 7.70/2.90 7.70/2.90 less_out_gg(x1, x2) = less_out_gg 7.70/2.90 7.70/2.90 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.90 7.70/2.90 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 7.70/2.90 7.70/2.90 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6) 7.70/2.90 7.70/2.90 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 7.70/2.90 7.70/2.90 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 7.70/2.90 7.70/2.90 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 7.70/2.90 7.70/2.90 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 7.70/2.90 7.70/2.90 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.70/2.90 7.70/2.90 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.70/2.90 7.70/2.90 U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5) 7.70/2.90 7.70/2.90 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 7.70/2.90 7.70/2.90 7.70/2.90 We have to consider all (P,R,Pi)-chains 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (15) UsableRulesProof (EQUIVALENT) 7.70/2.90 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (16) 7.70/2.90 Obligation: 7.70/2.90 Pi DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 7.70/2.90 7.70/2.90 R is empty. 7.70/2.90 Pi is empty. 7.70/2.90 We have to consider all (P,R,Pi)-chains 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (17) PiDPToQDPProof (EQUIVALENT) 7.70/2.90 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (18) 7.70/2.90 Obligation: 7.70/2.90 Q DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 7.70/2.90 7.70/2.90 R is empty. 7.70/2.90 Q is empty. 7.70/2.90 We have to consider all (P,Q,R)-chains. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (19) QDPSizeChangeProof (EQUIVALENT) 7.70/2.90 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. 7.70/2.90 7.70/2.90 From the DPs we obtained the following set of size-change graphs: 7.70/2.90 *LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 7.70/2.90 The graph contains the following edges 1 > 1, 2 > 2 7.70/2.90 7.70/2.90 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (20) 7.70/2.90 YES 7.70/2.90 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (21) 7.70/2.90 Obligation: 7.70/2.90 Pi DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 7.70/2.90 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.90 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 7.70/2.90 7.70/2.90 The TRS R consists of the following rules: 7.70/2.90 7.70/2.90 qs_in_ga([], []) -> qs_out_ga([], []) 7.70/2.90 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.90 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.90 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.90 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.90 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.90 U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.90 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.90 part_in_ggaa(X1, [], [], []) -> part_out_ggaa(X1, [], [], []) 7.70/2.90 U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 7.70/2.90 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 7.70/2.90 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.90 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.90 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.90 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.70/2.90 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.90 U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.70/2.90 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 7.70/2.90 7.70/2.90 The argument filtering Pi contains the following mapping: 7.70/2.90 qs_in_ga(x1, x2) = qs_in_ga(x1) 7.70/2.90 7.70/2.90 [] = [] 7.70/2.90 7.70/2.90 qs_out_ga(x1, x2) = qs_out_ga(x2) 7.70/2.90 7.70/2.90 .(x1, x2) = .(x1, x2) 7.70/2.90 7.70/2.90 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 7.70/2.90 7.70/2.90 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 7.70/2.90 7.70/2.90 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 7.70/2.90 7.70/2.90 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.90 7.70/2.90 0 = 0 7.70/2.90 7.70/2.90 s(x1) = s(x1) 7.70/2.90 7.70/2.90 less_out_gg(x1, x2) = less_out_gg 7.70/2.90 7.70/2.90 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.90 7.70/2.90 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 7.70/2.90 7.70/2.90 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6) 7.70/2.90 7.70/2.90 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 7.70/2.90 7.70/2.90 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 7.70/2.90 7.70/2.90 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 7.70/2.90 7.70/2.90 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 7.70/2.90 7.70/2.90 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.70/2.90 7.70/2.90 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.70/2.90 7.70/2.90 U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5) 7.70/2.90 7.70/2.90 PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) 7.70/2.90 7.70/2.90 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 7.70/2.90 7.70/2.90 7.70/2.90 We have to consider all (P,R,Pi)-chains 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (22) UsableRulesProof (EQUIVALENT) 7.70/2.90 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (23) 7.70/2.90 Obligation: 7.70/2.90 Pi DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 7.70/2.90 PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.90 PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GGAA(X, Xs, Ls, Bs) 7.70/2.90 7.70/2.90 The TRS R consists of the following rules: 7.70/2.90 7.70/2.90 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.90 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.90 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.90 7.70/2.90 The argument filtering Pi contains the following mapping: 7.70/2.90 .(x1, x2) = .(x1, x2) 7.70/2.90 7.70/2.90 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.90 7.70/2.90 0 = 0 7.70/2.90 7.70/2.90 s(x1) = s(x1) 7.70/2.90 7.70/2.90 less_out_gg(x1, x2) = less_out_gg 7.70/2.90 7.70/2.90 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.90 7.70/2.90 PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) 7.70/2.90 7.70/2.90 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 7.70/2.90 7.70/2.90 7.70/2.90 We have to consider all (P,R,Pi)-chains 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (24) PiDPToQDPProof (SOUND) 7.70/2.90 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (25) 7.70/2.90 Obligation: 7.70/2.90 Q DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 U5_GGAA(X, Y, Xs, less_out_gg) -> PART_IN_GGAA(X, Xs) 7.70/2.90 PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, less_in_gg(X, Y)) 7.70/2.90 PART_IN_GGAA(X, .(Y, Xs)) -> PART_IN_GGAA(X, Xs) 7.70/2.90 7.70/2.90 The TRS R consists of the following rules: 7.70/2.90 7.70/2.90 less_in_gg(0, s(X2)) -> less_out_gg 7.70/2.90 less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y)) 7.70/2.90 U9_gg(less_out_gg) -> less_out_gg 7.70/2.90 7.70/2.90 The set Q consists of the following terms: 7.70/2.90 7.70/2.90 less_in_gg(x0, x1) 7.70/2.90 U9_gg(x0) 7.70/2.90 7.70/2.90 We have to consider all (P,Q,R)-chains. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (26) QDPSizeChangeProof (EQUIVALENT) 7.70/2.90 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. 7.70/2.90 7.70/2.90 From the DPs we obtained the following set of size-change graphs: 7.70/2.90 *PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, less_in_gg(X, Y)) 7.70/2.90 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 7.70/2.90 7.70/2.90 7.70/2.90 *PART_IN_GGAA(X, .(Y, Xs)) -> PART_IN_GGAA(X, Xs) 7.70/2.90 The graph contains the following edges 1 >= 1, 2 > 2 7.70/2.90 7.70/2.90 7.70/2.90 *U5_GGAA(X, Y, Xs, less_out_gg) -> PART_IN_GGAA(X, Xs) 7.70/2.90 The graph contains the following edges 1 >= 1, 3 >= 2 7.70/2.90 7.70/2.90 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (27) 7.70/2.90 YES 7.70/2.90 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (28) 7.70/2.90 Obligation: 7.70/2.90 Pi DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.90 U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) 7.70/2.90 QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.90 U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) 7.70/2.90 7.70/2.90 The TRS R consists of the following rules: 7.70/2.90 7.70/2.90 qs_in_ga([], []) -> qs_out_ga([], []) 7.70/2.90 qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) 7.70/2.90 part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y)) 7.70/2.90 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 7.70/2.90 less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y)) 7.70/2.90 U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 7.70/2.90 U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.90 part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) 7.70/2.90 part_in_ggaa(X1, [], [], []) -> part_out_ggaa(X1, [], [], []) 7.70/2.90 U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) 7.70/2.90 U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) 7.70/2.90 U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) 7.70/2.90 U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) 7.70/2.90 U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) 7.70/2.90 app_in_gga([], X, X) -> app_out_gga([], X, X) 7.70/2.90 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 7.70/2.90 U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 7.70/2.90 U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) 7.70/2.90 7.70/2.90 The argument filtering Pi contains the following mapping: 7.70/2.90 qs_in_ga(x1, x2) = qs_in_ga(x1) 7.70/2.90 7.70/2.90 [] = [] 7.70/2.90 7.70/2.90 qs_out_ga(x1, x2) = qs_out_ga(x2) 7.70/2.90 7.70/2.90 .(x1, x2) = .(x1, x2) 7.70/2.90 7.70/2.90 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 7.70/2.90 7.70/2.90 part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) 7.70/2.90 7.70/2.90 U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) 7.70/2.90 7.70/2.90 less_in_gg(x1, x2) = less_in_gg(x1, x2) 7.70/2.90 7.70/2.90 0 = 0 7.70/2.90 7.70/2.90 s(x1) = s(x1) 7.70/2.90 7.70/2.90 less_out_gg(x1, x2) = less_out_gg 7.70/2.90 7.70/2.90 U9_gg(x1, x2, x3) = U9_gg(x3) 7.70/2.90 7.70/2.90 U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) 7.70/2.90 7.70/2.90 U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6) 7.70/2.90 7.70/2.90 part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) 7.70/2.90 7.70/2.90 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) 7.70/2.90 7.70/2.90 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) 7.70/2.90 7.70/2.90 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 7.70/2.90 7.70/2.90 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 7.70/2.90 7.70/2.90 app_out_gga(x1, x2, x3) = app_out_gga(x3) 7.70/2.90 7.70/2.90 U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5) 7.70/2.90 7.70/2.90 QS_IN_GA(x1, x2) = QS_IN_GA(x1) 7.70/2.90 7.70/2.90 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 7.70/2.90 7.70/2.90 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) 7.70/2.90 7.70/2.90 7.70/2.90 We have to consider all (P,R,Pi)-chains 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (29) PiDPToQDPProof (SOUND) 7.70/2.90 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.70/2.90 ---------------------------------------- 7.70/2.90 7.70/2.90 (30) 7.70/2.90 Obligation: 7.70/2.90 Q DP problem: 7.70/2.90 The TRS P consists of the following rules: 7.70/2.90 7.70/2.90 U1_GA(X, part_out_ggaa(Littles, Bigs)) -> U2_GA(X, Bigs, qs_in_ga(Littles)) 7.70/2.90 U2_GA(X, Bigs, qs_out_ga(Ls)) -> QS_IN_GA(Bigs) 7.70/2.90 QS_IN_GA(.(X, Xs)) -> U1_GA(X, part_in_ggaa(X, Xs)) 7.70/2.90 U1_GA(X, part_out_ggaa(Littles, Bigs)) -> QS_IN_GA(Littles) 7.70/2.91 7.70/2.91 The TRS R consists of the following rules: 7.70/2.91 7.70/2.91 qs_in_ga([]) -> qs_out_ga([]) 7.70/2.91 qs_in_ga(.(X, Xs)) -> U1_ga(X, part_in_ggaa(X, Xs)) 7.70/2.91 part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, less_in_gg(X, Y)) 7.70/2.91 less_in_gg(0, s(X2)) -> less_out_gg 7.70/2.91 less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y)) 7.70/2.91 U9_gg(less_out_gg) -> less_out_gg 7.70/2.91 U5_ggaa(X, Y, Xs, less_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) 7.70/2.91 part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(Y, part_in_ggaa(X, Xs)) 7.70/2.91 part_in_ggaa(X1, []) -> part_out_ggaa([], []) 7.70/2.91 U7_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) 7.70/2.91 U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) 7.70/2.91 U1_ga(X, part_out_ggaa(Littles, Bigs)) -> U2_ga(X, Bigs, qs_in_ga(Littles)) 7.70/2.91 U2_ga(X, Bigs, qs_out_ga(Ls)) -> U3_ga(X, Ls, qs_in_ga(Bigs)) 7.70/2.91 U3_ga(X, Ls, qs_out_ga(Bs)) -> U4_ga(app_in_gga(Ls, .(X, Bs))) 7.70/2.91 app_in_gga([], X) -> app_out_gga(X) 7.70/2.91 app_in_gga(.(X, Xs), Ys) -> U8_gga(X, app_in_gga(Xs, Ys)) 7.70/2.91 U8_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) 7.70/2.91 U4_ga(app_out_gga(Ys)) -> qs_out_ga(Ys) 7.70/2.91 7.70/2.91 The set Q consists of the following terms: 7.70/2.91 7.70/2.91 qs_in_ga(x0) 7.70/2.91 part_in_ggaa(x0, x1) 7.70/2.91 less_in_gg(x0, x1) 7.70/2.91 U9_gg(x0) 7.70/2.91 U5_ggaa(x0, x1, x2, x3) 7.70/2.91 U7_ggaa(x0, x1) 7.70/2.91 U6_ggaa(x0, x1) 7.70/2.91 U1_ga(x0, x1) 7.70/2.91 U2_ga(x0, x1, x2) 7.70/2.91 U3_ga(x0, x1, x2) 7.70/2.91 app_in_gga(x0, x1) 7.70/2.91 U8_gga(x0, x1) 7.70/2.91 U4_ga(x0) 7.70/2.91 7.70/2.91 We have to consider all (P,Q,R)-chains. 7.70/2.91 ---------------------------------------- 7.70/2.91 7.70/2.91 (31) QDPOrderProof (EQUIVALENT) 7.70/2.91 We use the reduction pair processor [LPAR04,JAR06]. 7.70/2.91 7.70/2.91 7.70/2.91 The following pairs can be oriented strictly and are deleted. 7.70/2.91 7.70/2.91 U1_GA(X, part_out_ggaa(Littles, Bigs)) -> U2_GA(X, Bigs, qs_in_ga(Littles)) 7.70/2.91 U1_GA(X, part_out_ggaa(Littles, Bigs)) -> QS_IN_GA(Littles) 7.70/2.91 The remaining pairs can at least be oriented weakly. 7.70/2.91 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 7.70/2.91 7.70/2.91 POL( U2_GA_3(x_1, ..., x_3) ) = x_2 + 1 7.70/2.91 POL( qs_in_ga_1(x_1) ) = 0 7.70/2.91 POL( [] ) = 0 7.70/2.91 POL( qs_out_ga_1(x_1) ) = max{0, x_1 - 2} 7.70/2.91 POL( ._2(x_1, x_2) ) = x_2 + 1 7.70/2.91 POL( U1_ga_2(x_1, x_2) ) = max{0, -2} 7.70/2.91 POL( part_in_ggaa_2(x_1, x_2) ) = x_2 + 2 7.70/2.91 POL( U1_GA_2(x_1, x_2) ) = x_2 7.70/2.91 POL( U5_ggaa_4(x_1, ..., x_4) ) = max{0, x_3 + 2x_4 - 1} 7.70/2.91 POL( less_in_gg_2(x_1, x_2) ) = 2 7.70/2.91 POL( U7_ggaa_2(x_1, x_2) ) = x_2 + 1 7.70/2.91 POL( part_out_ggaa_2(x_1, x_2) ) = x_1 + x_2 + 2 7.70/2.91 POL( U2_ga_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2 7.70/2.91 POL( U3_ga_3(x_1, ..., x_3) ) = 2x_2 7.70/2.91 POL( U4_ga_1(x_1) ) = 2 7.70/2.91 POL( app_in_gga_2(x_1, x_2) ) = max{0, x_2 - 2} 7.70/2.91 POL( less_out_gg ) = 2 7.70/2.91 POL( U6_ggaa_2(x_1, x_2) ) = x_2 + 1 7.70/2.91 POL( 0 ) = 2 7.70/2.91 POL( s_1(x_1) ) = 0 7.70/2.91 POL( U9_gg_1(x_1) ) = 2 7.70/2.91 POL( app_out_gga_1(x_1) ) = max{0, x_1 - 2} 7.70/2.91 POL( U8_gga_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} 7.70/2.91 POL( QS_IN_GA_1(x_1) ) = x_1 + 1 7.70/2.91 7.70/2.91 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 7.70/2.91 7.70/2.91 part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, less_in_gg(X, Y)) 7.70/2.91 part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(Y, part_in_ggaa(X, Xs)) 7.70/2.91 part_in_ggaa(X1, []) -> part_out_ggaa([], []) 7.70/2.91 U5_ggaa(X, Y, Xs, less_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) 7.70/2.91 U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) 7.70/2.91 less_in_gg(0, s(X2)) -> less_out_gg 7.70/2.91 less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y)) 7.70/2.91 U7_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) 7.70/2.91 U9_gg(less_out_gg) -> less_out_gg 7.70/2.91 7.70/2.91 7.70/2.91 ---------------------------------------- 7.70/2.91 7.70/2.91 (32) 7.70/2.91 Obligation: 7.70/2.91 Q DP problem: 7.70/2.91 The TRS P consists of the following rules: 7.70/2.91 7.70/2.91 U2_GA(X, Bigs, qs_out_ga(Ls)) -> QS_IN_GA(Bigs) 7.70/2.91 QS_IN_GA(.(X, Xs)) -> U1_GA(X, part_in_ggaa(X, Xs)) 7.70/2.91 7.70/2.91 The TRS R consists of the following rules: 7.70/2.91 7.70/2.91 qs_in_ga([]) -> qs_out_ga([]) 7.70/2.91 qs_in_ga(.(X, Xs)) -> U1_ga(X, part_in_ggaa(X, Xs)) 7.70/2.91 part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, less_in_gg(X, Y)) 7.70/2.91 less_in_gg(0, s(X2)) -> less_out_gg 7.70/2.91 less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y)) 7.70/2.91 U9_gg(less_out_gg) -> less_out_gg 7.70/2.91 U5_ggaa(X, Y, Xs, less_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) 7.70/2.91 part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(Y, part_in_ggaa(X, Xs)) 7.70/2.91 part_in_ggaa(X1, []) -> part_out_ggaa([], []) 7.70/2.91 U7_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) 7.70/2.91 U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) 7.70/2.91 U1_ga(X, part_out_ggaa(Littles, Bigs)) -> U2_ga(X, Bigs, qs_in_ga(Littles)) 7.70/2.91 U2_ga(X, Bigs, qs_out_ga(Ls)) -> U3_ga(X, Ls, qs_in_ga(Bigs)) 7.70/2.91 U3_ga(X, Ls, qs_out_ga(Bs)) -> U4_ga(app_in_gga(Ls, .(X, Bs))) 7.70/2.91 app_in_gga([], X) -> app_out_gga(X) 7.70/2.91 app_in_gga(.(X, Xs), Ys) -> U8_gga(X, app_in_gga(Xs, Ys)) 7.70/2.91 U8_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) 7.70/2.91 U4_ga(app_out_gga(Ys)) -> qs_out_ga(Ys) 7.70/2.91 7.70/2.91 The set Q consists of the following terms: 7.70/2.91 7.70/2.91 qs_in_ga(x0) 7.70/2.91 part_in_ggaa(x0, x1) 7.70/2.91 less_in_gg(x0, x1) 7.70/2.91 U9_gg(x0) 7.70/2.91 U5_ggaa(x0, x1, x2, x3) 7.70/2.91 U7_ggaa(x0, x1) 7.70/2.91 U6_ggaa(x0, x1) 7.70/2.91 U1_ga(x0, x1) 7.70/2.91 U2_ga(x0, x1, x2) 7.70/2.91 U3_ga(x0, x1, x2) 7.70/2.91 app_in_gga(x0, x1) 7.70/2.91 U8_gga(x0, x1) 7.70/2.91 U4_ga(x0) 7.70/2.91 7.70/2.91 We have to consider all (P,Q,R)-chains. 7.70/2.91 ---------------------------------------- 7.70/2.91 7.70/2.91 (33) DependencyGraphProof (EQUIVALENT) 7.70/2.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 7.70/2.91 ---------------------------------------- 7.70/2.91 7.70/2.91 (34) 7.70/2.91 TRUE 8.06/2.98 EOF