/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern minsort(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 5 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 4 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [EQUIVALENT, 5 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [EQUIVALENT, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [EQUIVALENT, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES (35) PiDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) PiDP (38) PiDPToQDPProof [SOUND, 0 ms] (39) QDP (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] (41) YES (42) PiDP (43) UsableRulesProof [EQUIVALENT, 0 ms] (44) PiDP (45) PiDPToQDPProof [SOUND, 0 ms] (46) QDP (47) QDPQMonotonicMRRProof [EQUIVALENT, 70 ms] (48) QDP (49) DependencyGraphProof [EQUIVALENT, 0 ms] (50) TRUE ---------------------------------------- (0) Obligation: Clauses: minsort([], []). minsort(L, .(X, L1)) :- ','(min1(X, L), ','(remove(X, L, L2), minsort(L2, L1))). min1(M, .(X, L)) :- min2(X, M, L). min2(X, X, []). min2(X, A, .(M, L)) :- ','(min(X, M, B), min2(B, A, L)). min(X, Y, X) :- le(X, Y). min(X, Y, Y) :- gt(X, Y). remove(N, .(N, L), L). remove(N, .(M, L), .(M, L1)) :- ','(notEq(N, M), remove(N, L, L1)). gt(s(X), s(Y)) :- gt(X, Y). gt(s(X), 0). le(s(X), s(Y)) :- le(X, Y). le(0, s(Y)). le(0, 0). notEq(s(X), s(Y)) :- notEq(X, Y). notEq(s(X), 0). notEq(0, s(X)). Query: minsort(g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: minsort_in_2: (b,f) min1_in_2: (f,b) min2_in_3: (b,f,b) min_in_3: (b,b,f) le_in_2: (b,b) gt_in_2: (b,b) remove_in_3: (b,b,f) notEq_in_2: (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: MINSORT_IN_GA(L, .(X, L1)) -> U1_GA(L, X, L1, min1_in_ag(X, L)) MINSORT_IN_GA(L, .(X, L1)) -> MIN1_IN_AG(X, L) MIN1_IN_AG(M, .(X, L)) -> U4_AG(M, X, L, min2_in_gag(X, M, L)) MIN1_IN_AG(M, .(X, L)) -> MIN2_IN_GAG(X, M, L) MIN2_IN_GAG(X, A, .(M, L)) -> U5_GAG(X, A, M, L, min_in_gga(X, M, B)) MIN2_IN_GAG(X, A, .(M, L)) -> MIN_IN_GGA(X, M, B) MIN_IN_GGA(X, Y, X) -> U7_GGA(X, Y, le_in_gg(X, Y)) MIN_IN_GGA(X, Y, X) -> LE_IN_GG(X, Y) LE_IN_GG(s(X), s(Y)) -> U12_GG(X, Y, le_in_gg(X, Y)) LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) MIN_IN_GGA(X, Y, Y) -> U8_GGA(X, Y, gt_in_gg(X, Y)) MIN_IN_GGA(X, Y, Y) -> GT_IN_GG(X, Y) GT_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, gt_in_gg(X, Y)) GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> U6_GAG(X, A, M, L, min2_in_gag(B, A, L)) U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> MIN2_IN_GAG(B, A, L) U1_GA(L, X, L1, min1_out_ag(X, L)) -> U2_GA(L, X, L1, remove_in_gga(X, L, L2)) U1_GA(L, X, L1, min1_out_ag(X, L)) -> REMOVE_IN_GGA(X, L, L2) REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> U9_GGA(N, M, L, L1, notEq_in_gg(N, M)) REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> NOTEQ_IN_GG(N, M) NOTEQ_IN_GG(s(X), s(Y)) -> U13_GG(X, Y, notEq_in_gg(X, Y)) NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> U10_GGA(N, M, L, L1, remove_in_gga(N, L, L1)) U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> REMOVE_IN_GGA(N, L, L1) U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> U3_GA(L, X, L1, minsort_in_ga(L2, L1)) U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> MINSORT_IN_GA(L2, L1) The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) MIN1_IN_AG(x1, x2) = MIN1_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x4) MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3) U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5) MIN_IN_GGA(x1, x2, x3) = MIN_IN_GGA(x1, x2) U7_GGA(x1, x2, x3) = U7_GGA(x1, x3) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) U12_GG(x1, x2, x3) = U12_GG(x3) U8_GGA(x1, x2, x3) = U8_GGA(x2, x3) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) U11_GG(x1, x2, x3) = U11_GG(x3) U6_GAG(x1, x2, x3, x4, x5) = U6_GAG(x5) U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5) NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2) U13_GG(x1, x2, x3) = U13_GG(x3) U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x2, x5) U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: MINSORT_IN_GA(L, .(X, L1)) -> U1_GA(L, X, L1, min1_in_ag(X, L)) MINSORT_IN_GA(L, .(X, L1)) -> MIN1_IN_AG(X, L) MIN1_IN_AG(M, .(X, L)) -> U4_AG(M, X, L, min2_in_gag(X, M, L)) MIN1_IN_AG(M, .(X, L)) -> MIN2_IN_GAG(X, M, L) MIN2_IN_GAG(X, A, .(M, L)) -> U5_GAG(X, A, M, L, min_in_gga(X, M, B)) MIN2_IN_GAG(X, A, .(M, L)) -> MIN_IN_GGA(X, M, B) MIN_IN_GGA(X, Y, X) -> U7_GGA(X, Y, le_in_gg(X, Y)) MIN_IN_GGA(X, Y, X) -> LE_IN_GG(X, Y) LE_IN_GG(s(X), s(Y)) -> U12_GG(X, Y, le_in_gg(X, Y)) LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) MIN_IN_GGA(X, Y, Y) -> U8_GGA(X, Y, gt_in_gg(X, Y)) MIN_IN_GGA(X, Y, Y) -> GT_IN_GG(X, Y) GT_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, gt_in_gg(X, Y)) GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> U6_GAG(X, A, M, L, min2_in_gag(B, A, L)) U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> MIN2_IN_GAG(B, A, L) U1_GA(L, X, L1, min1_out_ag(X, L)) -> U2_GA(L, X, L1, remove_in_gga(X, L, L2)) U1_GA(L, X, L1, min1_out_ag(X, L)) -> REMOVE_IN_GGA(X, L, L2) REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> U9_GGA(N, M, L, L1, notEq_in_gg(N, M)) REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> NOTEQ_IN_GG(N, M) NOTEQ_IN_GG(s(X), s(Y)) -> U13_GG(X, Y, notEq_in_gg(X, Y)) NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> U10_GGA(N, M, L, L1, remove_in_gga(N, L, L1)) U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> REMOVE_IN_GGA(N, L, L1) U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> U3_GA(L, X, L1, minsort_in_ga(L2, L1)) U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> MINSORT_IN_GA(L2, L1) The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) MIN1_IN_AG(x1, x2) = MIN1_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x4) MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3) U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5) MIN_IN_GGA(x1, x2, x3) = MIN_IN_GGA(x1, x2) U7_GGA(x1, x2, x3) = U7_GGA(x1, x3) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) U12_GG(x1, x2, x3) = U12_GG(x3) U8_GGA(x1, x2, x3) = U8_GGA(x2, x3) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) U11_GG(x1, x2, x3) = U11_GG(x3) U6_GAG(x1, x2, x3, x4, x5) = U6_GAG(x5) U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5) NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2) U13_GG(x1, x2, x3) = U13_GG(x3) U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x2, x5) U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 16 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> REMOVE_IN_GGA(N, L, L1) REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> U9_GGA(N, M, L, L1, notEq_in_gg(N, M)) The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) -> REMOVE_IN_GGA(N, L, L1) REMOVE_IN_GGA(N, .(M, L), .(M, L1)) -> U9_GGA(N, M, L, L1, notEq_in_gg(N, M)) The TRS R consists of the following rules: notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) s(x1) = s(x1) 0 = 0 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: U9_GGA(N, M, L, notEq_out_gg) -> REMOVE_IN_GGA(N, L) REMOVE_IN_GGA(N, .(M, L)) -> U9_GGA(N, M, L, notEq_in_gg(N, M)) The TRS R consists of the following rules: notEq_in_gg(s(X), s(Y)) -> U13_gg(notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg notEq_in_gg(0, s(X)) -> notEq_out_gg U13_gg(notEq_out_gg) -> notEq_out_gg The set Q consists of the following terms: notEq_in_gg(x0, x1) U13_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *REMOVE_IN_GGA(N, .(M, L)) -> U9_GGA(N, M, L, notEq_in_gg(N, M)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 *U9_GGA(N, M, L, notEq_out_gg) -> REMOVE_IN_GGA(N, L) The graph contains the following edges 1 >= 1, 3 >= 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> MIN2_IN_GAG(B, A, L) MIN2_IN_GAG(X, A, .(M, L)) -> U5_GAG(X, A, M, L, min_in_gga(X, M, B)) The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3) U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (37) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAG(X, A, M, L, min_out_gga(X, M, B)) -> MIN2_IN_GAG(B, A, L) MIN2_IN_GAG(X, A, .(M, L)) -> U5_GAG(X, A, M, L, min_in_gga(X, M, B)) The TRS R consists of the following rules: min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3) U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (38) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GAG(L, min_out_gga(B)) -> MIN2_IN_GAG(B, L) MIN2_IN_GAG(X, .(M, L)) -> U5_GAG(L, min_in_gga(X, M)) The TRS R consists of the following rules: min_in_gga(X, Y) -> U7_gga(X, le_in_gg(X, Y)) min_in_gga(X, Y) -> U8_gga(Y, gt_in_gg(X, Y)) U7_gga(X, le_out_gg) -> min_out_gga(X) U8_gga(Y, gt_out_gg) -> min_out_gga(Y) le_in_gg(s(X), s(Y)) -> U12_gg(le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg gt_in_gg(s(X), s(Y)) -> U11_gg(gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg U12_gg(le_out_gg) -> le_out_gg U11_gg(gt_out_gg) -> gt_out_gg The set Q consists of the following terms: min_in_gga(x0, x1) U7_gga(x0, x1) U8_gga(x0, x1) le_in_gg(x0, x1) gt_in_gg(x0, x1) U12_gg(x0) U11_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (40) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MIN2_IN_GAG(X, .(M, L)) -> U5_GAG(L, min_in_gga(X, M)) The graph contains the following edges 2 > 1 *U5_GAG(L, min_out_gga(B)) -> MIN2_IN_GAG(B, L) The graph contains the following edges 2 > 1, 1 >= 2 ---------------------------------------- (41) YES ---------------------------------------- (42) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(L, X, L1, min1_out_ag(X, L)) -> U2_GA(L, X, L1, remove_in_gga(X, L, L2)) U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> MINSORT_IN_GA(L2, L1) MINSORT_IN_GA(L, .(X, L1)) -> U1_GA(L, X, L1, min1_in_ag(X, L)) The TRS R consists of the following rules: minsort_in_ga([], []) -> minsort_out_ga([], []) minsort_in_ga(L, .(X, L1)) -> U1_ga(L, X, L1, min1_in_ag(X, L)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) U1_ga(L, X, L1, min1_out_ag(X, L)) -> U2_ga(L, X, L1, remove_in_gga(X, L, L2)) remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) U2_ga(L, X, L1, remove_out_gga(X, L, L2)) -> U3_ga(L, X, L1, minsort_in_ga(L2, L1)) U3_ga(L, X, L1, minsort_out_ga(L2, L1)) -> minsort_out_ga(L, .(X, L1)) The argument filtering Pi contains the following mapping: minsort_in_ga(x1, x2) = minsort_in_ga(x1) [] = [] minsort_out_ga(x1, x2) = minsort_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (43) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (44) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(L, X, L1, min1_out_ag(X, L)) -> U2_GA(L, X, L1, remove_in_gga(X, L, L2)) U2_GA(L, X, L1, remove_out_gga(X, L, L2)) -> MINSORT_IN_GA(L2, L1) MINSORT_IN_GA(L, .(X, L1)) -> U1_GA(L, X, L1, min1_in_ag(X, L)) The TRS R consists of the following rules: remove_in_gga(N, .(N, L), L) -> remove_out_gga(N, .(N, L), L) remove_in_gga(N, .(M, L), .(M, L1)) -> U9_gga(N, M, L, L1, notEq_in_gg(N, M)) min1_in_ag(M, .(X, L)) -> U4_ag(M, X, L, min2_in_gag(X, M, L)) U9_gga(N, M, L, L1, notEq_out_gg(N, M)) -> U10_gga(N, M, L, L1, remove_in_gga(N, L, L1)) U4_ag(M, X, L, min2_out_gag(X, M, L)) -> min1_out_ag(M, .(X, L)) notEq_in_gg(s(X), s(Y)) -> U13_gg(X, Y, notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) -> remove_out_gga(N, .(M, L), .(M, L1)) min2_in_gag(X, X, []) -> min2_out_gag(X, X, []) min2_in_gag(X, A, .(M, L)) -> U5_gag(X, A, M, L, min_in_gga(X, M, B)) U13_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U5_gag(X, A, M, L, min_out_gga(X, M, B)) -> U6_gag(X, A, M, L, min2_in_gag(B, A, L)) min_in_gga(X, Y, X) -> U7_gga(X, Y, le_in_gg(X, Y)) min_in_gga(X, Y, Y) -> U8_gga(X, Y, gt_in_gg(X, Y)) U6_gag(X, A, M, L, min2_out_gag(B, A, L)) -> min2_out_gag(X, A, .(M, L)) U7_gga(X, Y, le_out_gg(X, Y)) -> min_out_gga(X, Y, X) U8_gga(X, Y, gt_out_gg(X, Y)) -> min_out_gga(X, Y, Y) le_in_gg(s(X), s(Y)) -> U12_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) gt_in_gg(s(X), s(Y)) -> U11_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U12_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U11_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) The argument filtering Pi contains the following mapping: [] = [] min1_in_ag(x1, x2) = min1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3) min2_out_gag(x1, x2, x3) = min2_out_gag(x2) U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5) min_in_gga(x1, x2, x3) = min_in_gga(x1, x2) U7_gga(x1, x2, x3) = U7_gga(x1, x3) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U12_gg(x1, x2, x3) = U12_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg min_out_gga(x1, x2, x3) = min_out_gga(x3) U8_gga(x1, x2, x3) = U8_gga(x2, x3) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5) min1_out_ag(x1, x2) = min1_out_ag(x1) remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2) remove_out_gga(x1, x2, x3) = remove_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x3) notEq_out_gg(x1, x2) = notEq_out_gg U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5) MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (45) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(L, min1_out_ag(X)) -> U2_GA(X, remove_in_gga(X, L)) U2_GA(X, remove_out_gga(L2)) -> MINSORT_IN_GA(L2) MINSORT_IN_GA(L) -> U1_GA(L, min1_in_ag(L)) The TRS R consists of the following rules: remove_in_gga(N, .(N, L)) -> remove_out_gga(L) remove_in_gga(N, .(M, L)) -> U9_gga(N, M, L, notEq_in_gg(N, M)) min1_in_ag(.(X, L)) -> U4_ag(min2_in_gag(X, L)) U9_gga(N, M, L, notEq_out_gg) -> U10_gga(M, remove_in_gga(N, L)) U4_ag(min2_out_gag(M)) -> min1_out_ag(M) notEq_in_gg(s(X), s(Y)) -> U13_gg(notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg notEq_in_gg(0, s(X)) -> notEq_out_gg U10_gga(M, remove_out_gga(L1)) -> remove_out_gga(.(M, L1)) min2_in_gag(X, []) -> min2_out_gag(X) min2_in_gag(X, .(M, L)) -> U5_gag(L, min_in_gga(X, M)) U13_gg(notEq_out_gg) -> notEq_out_gg U5_gag(L, min_out_gga(B)) -> U6_gag(min2_in_gag(B, L)) min_in_gga(X, Y) -> U7_gga(X, le_in_gg(X, Y)) min_in_gga(X, Y) -> U8_gga(Y, gt_in_gg(X, Y)) U6_gag(min2_out_gag(A)) -> min2_out_gag(A) U7_gga(X, le_out_gg) -> min_out_gga(X) U8_gga(Y, gt_out_gg) -> min_out_gga(Y) le_in_gg(s(X), s(Y)) -> U12_gg(le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg gt_in_gg(s(X), s(Y)) -> U11_gg(gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg U12_gg(le_out_gg) -> le_out_gg U11_gg(gt_out_gg) -> gt_out_gg The set Q consists of the following terms: remove_in_gga(x0, x1) min1_in_ag(x0) U9_gga(x0, x1, x2, x3) U4_ag(x0) notEq_in_gg(x0, x1) U10_gga(x0, x1) min2_in_gag(x0, x1) U13_gg(x0) U5_gag(x0, x1) min_in_gga(x0, x1) U6_gag(x0) U7_gga(x0, x1) U8_gga(x0, x1) le_in_gg(x0, x1) gt_in_gg(x0, x1) U12_gg(x0) U11_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (47) QDPQMonotonicMRRProof (EQUIVALENT) 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. Strictly oriented dependency pairs: U1_GA(L, min1_out_ag(X)) -> U2_GA(X, remove_in_gga(X, L)) MINSORT_IN_GA(L) -> U1_GA(L, min1_in_ag(L)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 1 + x_2 POL(0) = 2 POL(MINSORT_IN_GA(x_1)) = 2 + 2*x_1 POL(U10_gga(x_1, x_2)) = 1 + x_2 POL(U11_gg(x_1)) = 2 POL(U12_gg(x_1)) = 2 POL(U13_gg(x_1)) = 2 POL(U1_GA(x_1, x_2)) = 1 + 2*x_1 POL(U2_GA(x_1, x_2)) = 2*x_2 POL(U4_ag(x_1)) = 2 POL(U5_gag(x_1, x_2)) = 0 POL(U6_gag(x_1)) = 0 POL(U7_gga(x_1, x_2)) = 2 POL(U8_gga(x_1, x_2)) = 2 POL(U9_gga(x_1, x_2, x_3, x_4)) = 1 + x_3 POL([]) = 2 POL(gt_in_gg(x_1, x_2)) = 2*x_1 POL(gt_out_gg) = 0 POL(le_in_gg(x_1, x_2)) = 2*x_2 POL(le_out_gg) = 2 POL(min1_in_ag(x_1)) = 1 + x_1 POL(min1_out_ag(x_1)) = 0 POL(min2_in_gag(x_1, x_2)) = 2 + x_1 POL(min2_out_gag(x_1)) = 0 POL(min_in_gga(x_1, x_2)) = 2 POL(min_out_gga(x_1)) = 0 POL(notEq_in_gg(x_1, x_2)) = 1 + x_2 POL(notEq_out_gg) = 0 POL(remove_in_gga(x_1, x_2)) = x_2 POL(remove_out_gga(x_1)) = 1 + x_1 POL(s(x_1)) = 2 + x_1 ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GA(X, remove_out_gga(L2)) -> MINSORT_IN_GA(L2) The TRS R consists of the following rules: remove_in_gga(N, .(N, L)) -> remove_out_gga(L) remove_in_gga(N, .(M, L)) -> U9_gga(N, M, L, notEq_in_gg(N, M)) min1_in_ag(.(X, L)) -> U4_ag(min2_in_gag(X, L)) U9_gga(N, M, L, notEq_out_gg) -> U10_gga(M, remove_in_gga(N, L)) U4_ag(min2_out_gag(M)) -> min1_out_ag(M) notEq_in_gg(s(X), s(Y)) -> U13_gg(notEq_in_gg(X, Y)) notEq_in_gg(s(X), 0) -> notEq_out_gg notEq_in_gg(0, s(X)) -> notEq_out_gg U10_gga(M, remove_out_gga(L1)) -> remove_out_gga(.(M, L1)) min2_in_gag(X, []) -> min2_out_gag(X) min2_in_gag(X, .(M, L)) -> U5_gag(L, min_in_gga(X, M)) U13_gg(notEq_out_gg) -> notEq_out_gg U5_gag(L, min_out_gga(B)) -> U6_gag(min2_in_gag(B, L)) min_in_gga(X, Y) -> U7_gga(X, le_in_gg(X, Y)) min_in_gga(X, Y) -> U8_gga(Y, gt_in_gg(X, Y)) U6_gag(min2_out_gag(A)) -> min2_out_gag(A) U7_gga(X, le_out_gg) -> min_out_gga(X) U8_gga(Y, gt_out_gg) -> min_out_gga(Y) le_in_gg(s(X), s(Y)) -> U12_gg(le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg gt_in_gg(s(X), s(Y)) -> U11_gg(gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg U12_gg(le_out_gg) -> le_out_gg U11_gg(gt_out_gg) -> gt_out_gg The set Q consists of the following terms: remove_in_gga(x0, x1) min1_in_ag(x0) U9_gga(x0, x1, x2, x3) U4_ag(x0) notEq_in_gg(x0, x1) U10_gga(x0, x1) min2_in_gag(x0, x1) U13_gg(x0) U5_gag(x0, x1) min_in_gga(x0, x1) U6_gag(x0) U7_gga(x0, x1) U8_gga(x0, x1) le_in_gg(x0, x1) gt_in_gg(x0, x1) U12_gg(x0) U11_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (49) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (50) TRUE