8.91/3.12 YES 9.19/3.19 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 9.19/3.19 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.19/3.19 9.19/3.19 9.19/3.19 Left Termination of the query pattern 9.19/3.19 9.19/3.19 minsort(g,a) 9.19/3.19 9.19/3.19 w.r.t. the given Prolog program could successfully be proven: 9.19/3.19 9.19/3.19 (0) Prolog 9.19/3.19 (1) PrologToPiTRSProof [SOUND, 0 ms] 9.19/3.19 (2) PiTRS 9.19/3.19 (3) DependencyPairsProof [EQUIVALENT, 3 ms] 9.19/3.19 (4) PiDP 9.19/3.19 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 9.19/3.19 (6) AND 9.19/3.19 (7) PiDP 9.19/3.19 (8) UsableRulesProof [EQUIVALENT, 0 ms] 9.19/3.19 (9) PiDP 9.19/3.19 (10) PiDPToQDPProof [EQUIVALENT, 12 ms] 9.19/3.19 (11) QDP 9.19/3.19 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.19/3.19 (13) YES 9.19/3.19 (14) PiDP 9.19/3.19 (15) UsableRulesProof [EQUIVALENT, 0 ms] 9.19/3.19 (16) PiDP 9.19/3.19 (17) PiDPToQDPProof [SOUND, 0 ms] 9.19/3.19 (18) QDP 9.19/3.19 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.19/3.19 (20) YES 9.19/3.19 (21) PiDP 9.19/3.19 (22) UsableRulesProof [EQUIVALENT, 0 ms] 9.19/3.19 (23) PiDP 9.19/3.19 (24) PiDPToQDPProof [EQUIVALENT, 0 ms] 9.19/3.19 (25) QDP 9.19/3.19 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.19/3.19 (27) YES 9.19/3.19 (28) PiDP 9.19/3.19 (29) UsableRulesProof [EQUIVALENT, 0 ms] 9.19/3.19 (30) PiDP 9.19/3.19 (31) PiDPToQDPProof [EQUIVALENT, 0 ms] 9.19/3.19 (32) QDP 9.19/3.19 (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.19/3.19 (34) YES 9.19/3.19 (35) PiDP 9.19/3.19 (36) UsableRulesProof [EQUIVALENT, 0 ms] 9.19/3.19 (37) PiDP 9.19/3.19 (38) PiDPToQDPProof [SOUND, 0 ms] 9.19/3.19 (39) QDP 9.19/3.19 (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.19/3.19 (41) YES 9.19/3.19 (42) PiDP 9.19/3.19 (43) UsableRulesProof [EQUIVALENT, 0 ms] 9.19/3.19 (44) PiDP 9.19/3.19 (45) PiDPToQDPProof [SOUND, 0 ms] 9.19/3.19 (46) QDP 9.19/3.19 (47) QDPQMonotonicMRRProof [EQUIVALENT, 55 ms] 9.19/3.19 (48) QDP 9.19/3.19 (49) DependencyGraphProof [EQUIVALENT, 0 ms] 9.19/3.19 (50) TRUE 9.19/3.19 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (0) 9.19/3.19 Obligation: 9.19/3.19 Clauses: 9.19/3.19 9.19/3.19 minsort([], []). 9.19/3.19 minsort(L, .(X, L1)) :- ','(min1(X, L), ','(remove(X, L, L2), minsort(L2, L1))). 9.19/3.19 min1(M, .(X, L)) :- min2(X, M, L). 9.19/3.19 min2(X, X, []). 9.19/3.19 min2(X, A, .(M, L)) :- ','(min(X, M, B), min2(B, A, L)). 9.19/3.19 min(X, Y, X) :- le(X, Y). 9.19/3.19 min(X, Y, Y) :- gt(X, Y). 9.19/3.19 remove(N, .(N, L), L). 9.19/3.19 remove(N, .(M, L), .(M, L1)) :- ','(notEq(N, M), remove(N, L, L1)). 9.19/3.19 gt(s(X), s(Y)) :- gt(X, Y). 9.19/3.19 gt(s(X), 0). 9.19/3.19 le(s(X), s(Y)) :- le(X, Y). 9.19/3.19 le(0, s(Y)). 9.19/3.19 le(0, 0). 9.19/3.19 notEq(s(X), s(Y)) :- notEq(X, Y). 9.19/3.19 notEq(s(X), 0). 9.19/3.19 notEq(0, s(X)). 9.19/3.19 9.19/3.19 9.19/3.19 Query: minsort(g,a) 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (1) PrologToPiTRSProof (SOUND) 9.19/3.19 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 9.19/3.19 9.19/3.19 minsort_in_2: (b,f) 9.19/3.19 9.19/3.19 min1_in_2: (f,b) 9.19/3.19 9.19/3.19 min2_in_3: (b,f,b) 9.19/3.19 9.19/3.19 min_in_3: (b,b,f) 9.19/3.19 9.19/3.19 le_in_2: (b,b) 9.19/3.19 9.19/3.19 gt_in_2: (b,b) 9.19/3.19 9.19/3.19 remove_in_3: (b,b,f) 9.19/3.19 9.19/3.19 notEq_in_2: (b,b) 9.19/3.19 9.19/3.19 Transforming Prolog into the following Term Rewriting System: 9.19/3.19 9.19/3.19 Pi-finite rewrite system: 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 9.19/3.19 9.19/3.19 9.19/3.19 9.19/3.19 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 9.19/3.19 9.19/3.19 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (2) 9.19/3.19 Obligation: 9.19/3.19 Pi-finite rewrite system: 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (3) DependencyPairsProof (EQUIVALENT) 9.19/3.19 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 MINSORT_IN_GA(L, .(X, L1)) -> U1_GA(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 MINSORT_IN_GA(L, .(X, L1)) -> MIN1_IN_AG(X, L) 9.19/3.19 MIN1_IN_AG(M, .(X, L)) -> U4_AG(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 MIN1_IN_AG(M, .(X, L)) -> MIN2_IN_GAG(X, M, L) 9.19/3.19 MIN2_IN_GAG(X, A, .(M, L)) -> U5_GAG(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 MIN2_IN_GAG(X, A, .(M, L)) -> MIN_IN_GGA(X, M, B) 9.19/3.19 MIN_IN_GGA(X, Y, X) -> U7_GGA(X, Y, le_in_gg(X, Y)) 9.19/3.19 MIN_IN_GGA(X, Y, X) -> LE_IN_GG(X, Y) 9.19/3.19 LE_IN_GG(s(X), s(Y)) -> U12_GG(X, Y, le_in_gg(X, Y)) 9.19/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.19/3.19 MIN_IN_GGA(X, Y, Y) -> U8_GGA(X, Y, gt_in_gg(X, Y)) 9.19/3.19 MIN_IN_GGA(X, Y, Y) -> GT_IN_GG(X, Y) 9.19/3.19 GT_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, gt_in_gg(X, Y)) 9.19/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.19/3.19 U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> U6_GAG(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> MIN2_IN_GAG(B, A, L) 9.19/3.19 U1_GA(L, X, L1, min1_out_ag(X, L)) -> U2_GA(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 U1_GA(L, X, L1, min1_out_ag(X, L)) -> REMOVE_IN_GGA(X, L, L2) 9.19/3.19 REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> U9_GGA(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> NOTEQ_IN_GG(N, M) 9.19/3.19 NOTEQ_IN_GG(s(X), s(Y)) -> U13_GG(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) 9.19/3.19 U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> U10_GGA(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> REMOVE_IN_GGA(N, L, L1) 9.19/3.19 U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> U3_GA(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> MINSORT_IN_GA(L2, L1) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1) 9.19/3.19 9.19/3.19 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 9.19/3.19 9.19/3.19 MIN1_IN_AG(x1, x2) = MIN1_IN_AG'(x2) 9.19/3.19 9.19/3.19 U4_AG(x1, x2, x3, x4) = U4_AG(x4) 9.19/3.19 9.19/3.19 MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3) 9.19/3.19 9.19/3.19 U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5) 9.19/3.19 9.19/3.19 MIN_IN_GGA(x1, x2, x3) = MIN_IN_GGA(x1, x2) 9.19/3.19 9.19/3.19 U7_GGA(x1, x2, x3) = U7_GGA(x1, x3) 9.19/3.19 9.19/3.19 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 U12_GG(x1, x2, x3) = U12_GG(x3) 9.19/3.19 9.19/3.19 U8_GGA(x1, x2, x3) = U8_GGA(x2, x3) 9.19/3.19 9.19/3.19 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 U11_GG(x1, x2, x3) = U11_GG(x3) 9.19/3.19 9.19/3.19 U6_GAG(x1, x2, x3, x4, x5) = U6_GAG(x5) 9.19/3.19 9.19/3.19 U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) 9.19/3.19 9.19/3.19 REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2) 9.19/3.19 9.19/3.19 U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 U13_GG(x1, x2, x3) = U13_GG(x3) 9.19/3.19 9.19/3.19 U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x2, x5) 9.19/3.19 9.19/3.19 U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (4) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 MINSORT_IN_GA(L, .(X, L1)) -> U1_GA(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 MINSORT_IN_GA(L, .(X, L1)) -> MIN1_IN_AG(X, L) 9.19/3.19 MIN1_IN_AG(M, .(X, L)) -> U4_AG(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 MIN1_IN_AG(M, .(X, L)) -> MIN2_IN_GAG(X, M, L) 9.19/3.19 MIN2_IN_GAG(X, A, .(M, L)) -> U5_GAG(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 MIN2_IN_GAG(X, A, .(M, L)) -> MIN_IN_GGA(X, M, B) 9.19/3.19 MIN_IN_GGA(X, Y, X) -> U7_GGA(X, Y, le_in_gg(X, Y)) 9.19/3.19 MIN_IN_GGA(X, Y, X) -> LE_IN_GG(X, Y) 9.19/3.19 LE_IN_GG(s(X), s(Y)) -> U12_GG(X, Y, le_in_gg(X, Y)) 9.19/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.19/3.19 MIN_IN_GGA(X, Y, Y) -> U8_GGA(X, Y, gt_in_gg(X, Y)) 9.19/3.19 MIN_IN_GGA(X, Y, Y) -> GT_IN_GG(X, Y) 9.19/3.19 GT_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, gt_in_gg(X, Y)) 9.19/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.19/3.19 U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> U6_GAG(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> MIN2_IN_GAG(B, A, L) 9.19/3.19 U1_GA(L, X, L1, min1_out_ag(X, L)) -> U2_GA(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 U1_GA(L, X, L1, min1_out_ag(X, L)) -> REMOVE_IN_GGA(X, L, L2) 9.19/3.19 REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> U9_GGA(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> NOTEQ_IN_GG(N, M) 9.19/3.19 NOTEQ_IN_GG(s(X), s(Y)) -> U13_GG(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) 9.19/3.19 U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> U10_GGA(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> REMOVE_IN_GGA(N, L, L1) 9.19/3.19 U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> U3_GA(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> MINSORT_IN_GA(L2, L1) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1) 9.19/3.19 9.19/3.19 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 9.19/3.19 9.19/3.19 MIN1_IN_AG(x1, x2) = MIN1_IN_AG(x2) 9.19/3.19 9.19/3.19 U4_AG(x1, x2, x3, x4) = U4_AG(x4) 9.19/3.19 9.19/3.19 MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3) 9.19/3.19 9.19/3.19 U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5) 9.19/3.19 9.19/3.19 MIN_IN_GGA(x1, x2, x3) = MIN_IN_GGA(x1, x2) 9.19/3.19 9.19/3.19 U7_GGA(x1, x2, x3) = U7_GGA(x1, x3) 9.19/3.19 9.19/3.19 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 U12_GG(x1, x2, x3) = U12_GG(x3) 9.19/3.19 9.19/3.19 U8_GGA(x1, x2, x3) = U8_GGA(x2, x3) 9.19/3.19 9.19/3.19 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 U11_GG(x1, x2, x3) = U11_GG(x3) 9.19/3.19 9.19/3.19 U6_GAG(x1, x2, x3, x4, x5) = U6_GAG(x5) 9.19/3.19 9.19/3.19 U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) 9.19/3.19 9.19/3.19 REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2) 9.19/3.19 9.19/3.19 U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 U13_GG(x1, x2, x3) = U13_GG(x3) 9.19/3.19 9.19/3.19 U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x2, x5) 9.19/3.19 9.19/3.19 U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (5) DependencyGraphProof (EQUIVALENT) 9.19/3.19 The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 16 less nodes. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (6) 9.19/3.19 Complex Obligation (AND) 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (7) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (8) UsableRulesProof (EQUIVALENT) 9.19/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (9) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) 9.19/3.19 9.19/3.19 R is empty. 9.19/3.19 Pi is empty. 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (10) PiDPToQDPProof (EQUIVALENT) 9.19/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (11) 9.19/3.19 Obligation: 9.19/3.19 Q DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) 9.19/3.19 9.19/3.19 R is empty. 9.19/3.19 Q is empty. 9.19/3.19 We have to consider all (P,Q,R)-chains. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (12) QDPSizeChangeProof (EQUIVALENT) 9.19/3.19 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. 9.19/3.19 9.19/3.19 From the DPs we obtained the following set of size-change graphs: 9.19/3.19 *NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) 9.19/3.19 The graph contains the following edges 1 > 1, 2 > 2 9.19/3.19 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (13) 9.19/3.19 YES 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (14) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> REMOVE_IN_GGA(N, L, L1) 9.19/3.19 REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> U9_GGA(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2) 9.19/3.19 9.19/3.19 U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (15) UsableRulesProof (EQUIVALENT) 9.19/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (16) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> REMOVE_IN_GGA(N, L, L1) 9.19/3.19 REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> U9_GGA(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2) 9.19/3.19 9.19/3.19 U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (17) PiDPToQDPProof (SOUND) 9.19/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (18) 9.19/3.19 Obligation: 9.19/3.19 Q DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U9_GGA(N, M, L, notEq_out_gg) -> REMOVE_IN_GGA(N, L) 9.19/3.19 REMOVE_IN_GGA(N, .(M, L)) -> U9_GGA(N, M, L, notEq_in_gg(N, M)) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg 9.19/3.19 U13_gg(notEq_out_gg) -> notEq_out_gg 9.19/3.19 9.19/3.19 The set Q consists of the following terms: 9.19/3.19 9.19/3.19 notEq_in_gg(x0, x1) 9.19/3.19 U13_gg(x0) 9.19/3.19 9.19/3.19 We have to consider all (P,Q,R)-chains. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (19) QDPSizeChangeProof (EQUIVALENT) 9.19/3.19 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. 9.19/3.19 9.19/3.19 From the DPs we obtained the following set of size-change graphs: 9.19/3.19 *REMOVE_IN_GGA(N, .(M, L)) -> U9_GGA(N, M, L, notEq_in_gg(N, M)) 9.19/3.19 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 9.19/3.19 9.19/3.19 9.19/3.19 *U9_GGA(N, M, L, notEq_out_gg) -> REMOVE_IN_GGA(N, L) 9.19/3.19 The graph contains the following edges 1 >= 1, 3 >= 2 9.19/3.19 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (20) 9.19/3.19 YES 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (21) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (22) UsableRulesProof (EQUIVALENT) 9.19/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (23) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.19/3.19 9.19/3.19 R is empty. 9.19/3.19 Pi is empty. 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (24) PiDPToQDPProof (EQUIVALENT) 9.19/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (25) 9.19/3.19 Obligation: 9.19/3.19 Q DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.19/3.19 9.19/3.19 R is empty. 9.19/3.19 Q is empty. 9.19/3.19 We have to consider all (P,Q,R)-chains. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (26) QDPSizeChangeProof (EQUIVALENT) 9.19/3.19 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. 9.19/3.19 9.19/3.19 From the DPs we obtained the following set of size-change graphs: 9.19/3.19 *GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 9.19/3.19 The graph contains the following edges 1 > 1, 2 > 2 9.19/3.19 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (27) 9.19/3.19 YES 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (28) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (29) UsableRulesProof (EQUIVALENT) 9.19/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (30) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.19/3.19 9.19/3.19 R is empty. 9.19/3.19 Pi is empty. 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (31) PiDPToQDPProof (EQUIVALENT) 9.19/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (32) 9.19/3.19 Obligation: 9.19/3.19 Q DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.19/3.19 9.19/3.19 R is empty. 9.19/3.19 Q is empty. 9.19/3.19 We have to consider all (P,Q,R)-chains. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (33) QDPSizeChangeProof (EQUIVALENT) 9.19/3.19 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. 9.19/3.19 9.19/3.19 From the DPs we obtained the following set of size-change graphs: 9.19/3.19 *LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 9.19/3.19 The graph contains the following edges 1 > 1, 2 > 2 9.19/3.19 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (34) 9.19/3.19 YES 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (35) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> MIN2_IN_GAG(B, A, L) 9.19/3.19 MIN2_IN_GAG(X, A, .(M, L)) -> U5_GAG(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3) 9.19/3.19 9.19/3.19 U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (36) UsableRulesProof (EQUIVALENT) 9.19/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (37) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> MIN2_IN_GAG(B, A, L) 9.19/3.19 MIN2_IN_GAG(X, A, .(M, L)) -> U5_GAG(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3) 9.19/3.19 9.19/3.19 U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (38) PiDPToQDPProof (SOUND) 9.19/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (39) 9.19/3.19 Obligation: 9.19/3.19 Q DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U5_GAG(L, min_out_gga(B)) -> MIN2_IN_GAG(B, L) 9.19/3.19 MIN2_IN_GAG(X, .(M, L)) -> U5_GAG(L, min_in_gga(X, M)) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 min_in_gga(X, Y) -> U7_gga(X, le_in_gg(X, Y)) 9.19/3.19 min_in_gga(X, Y) -> U8_gga(Y, gt_in_gg(X, Y)) 9.19/3.19 U7_gga(X, le_out_gg) -> min_out_gga(X) 9.19/3.19 U8_gga(Y, gt_out_gg) -> min_out_gga(Y) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg 9.19/3.19 le_in_gg(0, 0) -> le_out_gg 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg 9.19/3.19 U12_gg(le_out_gg) -> le_out_gg 9.19/3.19 U11_gg(gt_out_gg) -> gt_out_gg 9.19/3.19 9.19/3.19 The set Q consists of the following terms: 9.19/3.19 9.19/3.19 min_in_gga(x0, x1) 9.19/3.19 U7_gga(x0, x1) 9.19/3.19 U8_gga(x0, x1) 9.19/3.19 le_in_gg(x0, x1) 9.19/3.19 gt_in_gg(x0, x1) 9.19/3.19 U12_gg(x0) 9.19/3.19 U11_gg(x0) 9.19/3.19 9.19/3.19 We have to consider all (P,Q,R)-chains. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (40) QDPSizeChangeProof (EQUIVALENT) 9.19/3.19 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. 9.19/3.19 9.19/3.19 From the DPs we obtained the following set of size-change graphs: 9.19/3.19 *MIN2_IN_GAG(X, .(M, L)) -> U5_GAG(L, min_in_gga(X, M)) 9.19/3.19 The graph contains the following edges 2 > 1 9.19/3.19 9.19/3.19 9.19/3.19 *U5_GAG(L, min_out_gga(B)) -> MIN2_IN_GAG(B, L) 9.19/3.19 The graph contains the following edges 2 > 1, 1 >= 2 9.19/3.19 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (41) 9.19/3.19 YES 9.19/3.19 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (42) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U1_GA(L, X, L1, min1_out_ag(X, L)) -> U2_GA(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> MINSORT_IN_GA(L2, L1) 9.19/3.19 MINSORT_IN_GA(L, .(X, L1)) -> U1_GA(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 minsort_in_ga([], []) -> minsort_out_ga([], []) 9.19/3.19 minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) 9.19/3.19 U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 minsort_in_ga(x1, x2) = minsort_in_ga(x1) 9.19/3.19 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 minsort_out_ga(x1, x2) = minsort_out_ga(x2) 9.19/3.19 9.19/3.19 U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) 9.19/3.19 9.19/3.19 MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1) 9.19/3.19 9.19/3.19 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 9.19/3.19 9.19/3.19 U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (43) UsableRulesProof (EQUIVALENT) 9.19/3.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (44) 9.19/3.19 Obligation: 9.19/3.19 Pi DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U1_GA(L, X, L1, min1_out_ag(X, L)) -> U2_GA(L, X, L1, remove_in_gga(X, L, L2)) 9.19/3.19 U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> MINSORT_IN_GA(L2, L1) 9.19/3.19 MINSORT_IN_GA(L, .(X, L1)) -> U1_GA(L, X, L1, min1_in_ag(X, L)) 9.19/3.19 9.19/3.19 The TRS R consists of the following rules: 9.19/3.19 9.19/3.19 remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) 9.19/3.19 remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) 9.19/3.19 min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) 9.19/3.19 U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) 9.19/3.19 U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) 9.19/3.19 notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) 9.19/3.19 notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) 9.19/3.19 notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) 9.19/3.19 U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) 9.19/3.19 min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) 9.19/3.19 min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) 9.19/3.19 U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) 9.19/3.19 U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) 9.19/3.19 min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) 9.19/3.19 min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) 9.19/3.19 U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) 9.19/3.19 U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) 9.19/3.19 U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) 9.19/3.19 le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) 9.19/3.19 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 9.19/3.19 le_in_gg(0, 0) -> le_out_gg(0, 0) 9.19/3.19 gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) 9.19/3.19 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 9.19/3.19 U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 9.19/3.19 U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 9.19/3.19 9.19/3.19 The argument filtering Pi contains the following mapping: 9.19/3.19 [] = [] 9.19/3.19 9.19/3.19 min1_in_ag(x1, x2) = min1_in_ag(x2) 9.19/3.19 9.19/3.19 .(x1, x2) = .(x1, x2) 9.19/3.19 9.19/3.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 9.19/3.19 9.19/3.19 min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) 9.19/3.19 9.19/3.19 min2_out_gag(x1, x2, x3) = min2_out_gag(x2) 9.19/3.19 9.19/3.19 U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) 9.19/3.19 9.19/3.19 min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) 9.19/3.19 9.19/3.19 U7_gga(x1, x2, x3) = U7_gga(x1, x3) 9.19/3.19 9.19/3.19 le_in_gg(x1, x2) = le_in_gg(x1, x2) 9.19/3.19 9.19/3.19 s(x1) = s(x1) 9.19/3.19 9.19/3.19 U12_gg(x1, x2, x3) = U12_gg(x3) 9.19/3.19 9.19/3.19 0 = 0 9.19/3.19 9.19/3.19 le_out_gg(x1, x2) = le_out_gg 9.19/3.19 9.19/3.19 min_out_gga(x1, x2, x3) = min_out_gga(x3) 9.19/3.19 9.19/3.19 U8_gga(x1, x2, x3) = U8_gga(x2, x3) 9.19/3.19 9.19/3.19 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U11_gg(x1, x2, x3) = U11_gg(x3) 9.19/3.19 9.19/3.19 gt_out_gg(x1, x2) = gt_out_gg 9.19/3.19 9.19/3.19 U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) 9.19/3.19 9.19/3.19 min1_out_ag(x1, x2) = min1_out_ag(x1) 9.19/3.19 9.19/3.19 remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) 9.19/3.19 9.19/3.19 remove_out_gga(x1, x2, x3) = remove_out_gga(x3) 9.19/3.19 9.19/3.19 U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) 9.19/3.19 9.19/3.19 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) 9.19/3.19 9.19/3.19 U13_gg(x1, x2, x3) = U13_gg(x3) 9.19/3.19 9.19/3.19 notEq_out_gg(x1, x2) = notEq_out_gg 9.19/3.19 9.19/3.19 U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) 9.19/3.19 9.19/3.19 MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1) 9.19/3.19 9.19/3.19 U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) 9.19/3.19 9.19/3.19 U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) 9.19/3.19 9.19/3.19 9.19/3.19 We have to consider all (P,R,Pi)-chains 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (45) PiDPToQDPProof (SOUND) 9.19/3.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.19/3.19 ---------------------------------------- 9.19/3.19 9.19/3.19 (46) 9.19/3.19 Obligation: 9.19/3.19 Q DP problem: 9.19/3.19 The TRS P consists of the following rules: 9.19/3.19 9.19/3.19 U1_GA(L, min1_out_ag(X)) -> U2_GA(X, remove_in_gga(X, L)) 9.19/3.19 U2_GA(X, remove_out_gga(L2)) -> MINSORT_IN_GA(L2) 9.19/3.19 MINSORT_IN_GA(L) -> U1_GA(L, min1_in_ag(L)) 9.19/3.20 9.19/3.20 The TRS R consists of the following rules: 9.19/3.20 9.19/3.20 remove_in_gga(N, .(N, L)) -> remove_out_gga(L) 9.19/3.20 remove_in_gga(N, .(M, L)) -> U9_gga(N, M, L, notEq_in_gg(N, M)) 9.19/3.20 min1_in_ag(.(X, L)) -> U4_ag(min2_in_gag(X, L)) 9.19/3.20 U9_gga(N, M, L, notEq_out_gg) -> U10_gga(M, remove_in_gga(N, L)) 9.19/3.20 U4_ag(min2_out_gag(M)) -> min1_out_ag(M) 9.19/3.20 notEq_in_gg(s(X), s(Y)) -> U13_gg(notEq_in_gg(X, Y)) 9.19/3.20 notEq_in_gg(s(X), 0) -> notEq_out_gg 9.19/3.20 notEq_in_gg(0, s(X)) -> notEq_out_gg 9.19/3.20 U10_gga(M, remove_out_gga(L1)) -> remove_out_gga(.(M, L1)) 9.19/3.20 min2_in_gag(X, []) -> min2_out_gag(X) 9.19/3.20 min2_in_gag(X, .(M, L)) -> U5_gag(L, min_in_gga(X, M)) 9.19/3.20 U13_gg(notEq_out_gg) -> notEq_out_gg 9.19/3.20 U5_gag(L, min_out_gga(B)) -> U6_gag(min2_in_gag(B, L)) 9.19/3.20 min_in_gga(X, Y) -> U7_gga(X, le_in_gg(X, Y)) 9.19/3.20 min_in_gga(X, Y) -> U8_gga(Y, gt_in_gg(X, Y)) 9.19/3.20 U6_gag(min2_out_gag(A)) -> min2_out_gag(A) 9.19/3.20 U7_gga(X, le_out_gg) -> min_out_gga(X) 9.19/3.20 U8_gga(Y, gt_out_gg) -> min_out_gga(Y) 9.19/3.20 le_in_gg(s(X), s(Y)) -> U12_gg(le_in_gg(X, Y)) 9.19/3.20 le_in_gg(0, s(Y)) -> le_out_gg 9.19/3.20 le_in_gg(0, 0) -> le_out_gg 9.19/3.20 gt_in_gg(s(X), s(Y)) -> U11_gg(gt_in_gg(X, Y)) 9.19/3.20 gt_in_gg(s(X), 0) -> gt_out_gg 9.19/3.20 U12_gg(le_out_gg) -> le_out_gg 9.19/3.20 U11_gg(gt_out_gg) -> gt_out_gg 9.19/3.20 9.19/3.20 The set Q consists of the following terms: 9.19/3.20 9.19/3.20 remove_in_gga(x0, x1) 9.19/3.20 min1_in_ag(x0) 9.19/3.20 U9_gga(x0, x1, x2, x3) 9.19/3.20 U4_ag(x0) 9.19/3.20 notEq_in_gg(x0, x1) 9.19/3.20 U10_gga(x0, x1) 9.19/3.20 min2_in_gag(x0, x1) 9.19/3.20 U13_gg(x0) 9.19/3.20 U5_gag(x0, x1) 9.19/3.20 min_in_gga(x0, x1) 9.19/3.20 U6_gag(x0) 9.19/3.20 U7_gga(x0, x1) 9.19/3.20 U8_gga(x0, x1) 9.19/3.20 le_in_gg(x0, x1) 9.19/3.20 gt_in_gg(x0, x1) 9.19/3.20 U12_gg(x0) 9.19/3.20 U11_gg(x0) 9.19/3.20 9.19/3.20 We have to consider all (P,Q,R)-chains. 9.19/3.20 ---------------------------------------- 9.19/3.20 9.19/3.20 (47) QDPQMonotonicMRRProof (EQUIVALENT) 9.19/3.20 By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. 9.19/3.20 9.19/3.20 Strictly oriented dependency pairs: 9.19/3.20 9.19/3.20 U1_GA(L, min1_out_ag(X)) -> U2_GA(X, remove_in_gga(X, L)) 9.19/3.20 MINSORT_IN_GA(L) -> U1_GA(L, min1_in_ag(L)) 9.19/3.20 9.19/3.20 9.19/3.20 Used ordering: Polynomial interpretation [POLO]: 9.19/3.20 9.19/3.20 POL(.(x_1, x_2)) = 1 + x_2 9.19/3.20 POL(0) = 2 9.19/3.20 POL(MINSORT_IN_GA(x_1)) = 2 + 2*x_1 9.19/3.20 POL(U10_gga(x_1, x_2)) = 1 + x_2 9.19/3.20 POL(U11_gg(x_1)) = 2 9.19/3.20 POL(U12_gg(x_1)) = 2 9.19/3.20 POL(U13_gg(x_1)) = 2 9.19/3.20 POL(U1_GA(x_1, x_2)) = 1 + 2*x_1 9.19/3.20 POL(U2_GA(x_1, x_2)) = 2*x_2 9.19/3.20 POL(U4_ag(x_1)) = 2 9.19/3.20 POL(U5_gag(x_1, x_2)) = 0 9.19/3.20 POL(U6_gag(x_1)) = 0 9.19/3.20 POL(U7_gga(x_1, x_2)) = 2 9.19/3.20 POL(U8_gga(x_1, x_2)) = 2 9.19/3.20 POL(U9_gga(x_1, x_2, x_3, x_4)) = 1 + x_3 9.19/3.20 POL([]) = 2 9.19/3.20 POL(gt_in_gg(x_1, x_2)) = 2*x_1 9.19/3.20 POL(gt_out_gg) = 0 9.19/3.20 POL(le_in_gg(x_1, x_2)) = 2*x_2 9.19/3.20 POL(le_out_gg) = 2 9.19/3.20 POL(min1_in_ag(x_1)) = 1 + x_1 9.19/3.20 POL(min1_out_ag(x_1)) = 0 9.19/3.20 POL(min2_in_gag(x_1, x_2)) = 2 + x_1 9.19/3.20 POL(min2_out_gag(x_1)) = 0 9.19/3.20 POL(min_in_gga(x_1, x_2)) = 2 9.19/3.20 POL(min_out_gga(x_1)) = 0 9.19/3.20 POL(notEq_in_gg(x_1, x_2)) = 1 + x_2 9.19/3.20 POL(notEq_out_gg) = 0 9.19/3.20 POL(remove_in_gga(x_1, x_2)) = x_2 9.19/3.20 POL(remove_out_gga(x_1)) = 1 + x_1 9.19/3.20 POL(s(x_1)) = 2 + x_1 9.19/3.20 9.19/3.20 9.19/3.20 ---------------------------------------- 9.19/3.20 9.19/3.20 (48) 9.19/3.20 Obligation: 9.19/3.20 Q DP problem: 9.19/3.20 The TRS P consists of the following rules: 9.19/3.20 9.19/3.20 U2_GA(X, remove_out_gga(L2)) -> MINSORT_IN_GA(L2) 9.19/3.20 9.19/3.20 The TRS R consists of the following rules: 9.19/3.20 9.19/3.20 remove_in_gga(N, .(N, L)) -> remove_out_gga(L) 9.19/3.20 remove_in_gga(N, .(M, L)) -> U9_gga(N, M, L, notEq_in_gg(N, M)) 9.19/3.20 min1_in_ag(.(X, L)) -> U4_ag(min2_in_gag(X, L)) 9.19/3.20 U9_gga(N, M, L, notEq_out_gg) -> U10_gga(M, remove_in_gga(N, L)) 9.19/3.20 U4_ag(min2_out_gag(M)) -> min1_out_ag(M) 9.19/3.20 notEq_in_gg(s(X), s(Y)) -> U13_gg(notEq_in_gg(X, Y)) 9.19/3.20 notEq_in_gg(s(X), 0) -> notEq_out_gg 9.19/3.20 notEq_in_gg(0, s(X)) -> notEq_out_gg 9.19/3.20 U10_gga(M, remove_out_gga(L1)) -> remove_out_gga(.(M, L1)) 9.19/3.20 min2_in_gag(X, []) -> min2_out_gag(X) 9.19/3.20 min2_in_gag(X, .(M, L)) -> U5_gag(L, min_in_gga(X, M)) 9.19/3.20 U13_gg(notEq_out_gg) -> notEq_out_gg 9.19/3.20 U5_gag(L, min_out_gga(B)) -> U6_gag(min2_in_gag(B, L)) 9.19/3.20 min_in_gga(X, Y) -> U7_gga(X, le_in_gg(X, Y)) 9.19/3.20 min_in_gga(X, Y) -> U8_gga(Y, gt_in_gg(X, Y)) 9.19/3.20 U6_gag(min2_out_gag(A)) -> min2_out_gag(A) 9.19/3.20 U7_gga(X, le_out_gg) -> min_out_gga(X) 9.19/3.20 U8_gga(Y, gt_out_gg) -> min_out_gga(Y) 9.19/3.20 le_in_gg(s(X), s(Y)) -> U12_gg(le_in_gg(X, Y)) 9.19/3.20 le_in_gg(0, s(Y)) -> le_out_gg 9.19/3.20 le_in_gg(0, 0) -> le_out_gg 9.19/3.20 gt_in_gg(s(X), s(Y)) -> U11_gg(gt_in_gg(X, Y)) 9.19/3.20 gt_in_gg(s(X), 0) -> gt_out_gg 9.19/3.20 U12_gg(le_out_gg) -> le_out_gg 9.19/3.20 U11_gg(gt_out_gg) -> gt_out_gg 9.19/3.20 9.19/3.20 The set Q consists of the following terms: 9.19/3.20 9.19/3.20 remove_in_gga(x0, x1) 9.19/3.20 min1_in_ag(x0) 9.19/3.20 U9_gga(x0, x1, x2, x3) 9.19/3.20 U4_ag(x0) 9.19/3.20 notEq_in_gg(x0, x1) 9.19/3.20 U10_gga(x0, x1) 9.19/3.20 min2_in_gag(x0, x1) 9.19/3.20 U13_gg(x0) 9.19/3.20 U5_gag(x0, x1) 9.19/3.20 min_in_gga(x0, x1) 9.19/3.20 U6_gag(x0) 9.19/3.20 U7_gga(x0, x1) 9.19/3.20 U8_gga(x0, x1) 9.19/3.20 le_in_gg(x0, x1) 9.19/3.20 gt_in_gg(x0, x1) 9.19/3.20 U12_gg(x0) 9.19/3.20 U11_gg(x0) 9.19/3.20 9.19/3.20 We have to consider all (P,Q,R)-chains. 9.19/3.20 ---------------------------------------- 9.19/3.20 9.19/3.20 (49) DependencyGraphProof (EQUIVALENT) 9.19/3.20 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 9.19/3.20 ---------------------------------------- 9.19/3.20 9.19/3.20 (50) 9.19/3.20 TRUE 9.23/3.26 EOF