/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern div(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND,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, 6 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) QDPOrderProof [EQUIVALENT, 16 ms] (27) QDP (28) DependencyGraphProof [EQUIVALENT, 0 ms] (29) TRUE ---------------------------------------- (0) Obligation: Clauses: div(X, s(Y), Z) :- div_s(X, Y, Z). div_s(0, Y, 0). div_s(s(X), Y, 0) :- lss(X, Y). div_s(s(X), Y, s(Z)) :- ','(sub(X, Y, R), div_s(R, Y, Z)). lss(s(X), s(Y)) :- lss(X, Y). lss(0, s(Y)). sub(s(X), s(Y), Z) :- sub(X, Y, Z). sub(X, 0, X). Query: div(g,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: div_in_3: (b,b,f) div_s_in_3: (b,b,f) lss_in_2: (b,b) sub_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z)) div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0) div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y)) lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y)) lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y)) U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y)) U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0) div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R)) sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z)) sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X) U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z) U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z)) U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z)) U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x4) div_s_in_gga(x1, x2, x3) = div_s_in_gga(x1, x2) 0 = 0 div_s_out_gga(x1, x2, x3) = div_s_out_gga(x3) U2_gga(x1, x2, x3) = U2_gga(x3) lss_in_gg(x1, x2) = lss_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) lss_out_gg(x1, x2) = lss_out_gg U3_gga(x1, x2, x3, x4) = U3_gga(x2, x4) sub_in_gga(x1, x2, x3) = sub_in_gga(x1, x2) U6_gga(x1, x2, x3, x4) = U6_gga(x4) sub_out_gga(x1, x2, x3) = sub_out_gga(x3) U4_gga(x1, x2, x3, x4) = U4_gga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z)) div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0) div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y)) lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y)) lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y)) U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y)) U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0) div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R)) sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z)) sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X) U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z) U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z)) U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z)) U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x4) div_s_in_gga(x1, x2, x3) = div_s_in_gga(x1, x2) 0 = 0 div_s_out_gga(x1, x2, x3) = div_s_out_gga(x3) U2_gga(x1, x2, x3) = U2_gga(x3) lss_in_gg(x1, x2) = lss_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) lss_out_gg(x1, x2) = lss_out_gg U3_gga(x1, x2, x3, x4) = U3_gga(x2, x4) sub_in_gga(x1, x2, x3) = sub_in_gga(x1, x2) U6_gga(x1, x2, x3, x4) = U6_gga(x4) sub_out_gga(x1, x2, x3) = sub_out_gga(x3) U4_gga(x1, x2, x3, x4) = U4_gga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) ---------------------------------------- (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: DIV_IN_GGA(X, s(Y), Z) -> U1_GGA(X, Y, Z, div_s_in_gga(X, Y, Z)) DIV_IN_GGA(X, s(Y), Z) -> DIV_S_IN_GGA(X, Y, Z) DIV_S_IN_GGA(s(X), Y, 0) -> U2_GGA(X, Y, lss_in_gg(X, Y)) DIV_S_IN_GGA(s(X), Y, 0) -> LSS_IN_GG(X, Y) LSS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, lss_in_gg(X, Y)) LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y) DIV_S_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, sub_in_gga(X, Y, R)) DIV_S_IN_GGA(s(X), Y, s(Z)) -> SUB_IN_GGA(X, Y, R) SUB_IN_GGA(s(X), s(Y), Z) -> U6_GGA(X, Y, Z, sub_in_gga(X, Y, Z)) SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z) U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_GGA(X, Y, Z, div_s_in_gga(R, Y, Z)) U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> DIV_S_IN_GGA(R, Y, Z) The TRS R consists of the following rules: div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z)) div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0) div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y)) lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y)) lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y)) U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y)) U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0) div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R)) sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z)) sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X) U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z) U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z)) U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z)) U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x4) div_s_in_gga(x1, x2, x3) = div_s_in_gga(x1, x2) 0 = 0 div_s_out_gga(x1, x2, x3) = div_s_out_gga(x3) U2_gga(x1, x2, x3) = U2_gga(x3) lss_in_gg(x1, x2) = lss_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) lss_out_gg(x1, x2) = lss_out_gg U3_gga(x1, x2, x3, x4) = U3_gga(x2, x4) sub_in_gga(x1, x2, x3) = sub_in_gga(x1, x2) U6_gga(x1, x2, x3, x4) = U6_gga(x4) sub_out_gga(x1, x2, x3) = sub_out_gga(x3) U4_gga(x1, x2, x3, x4) = U4_gga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) DIV_S_IN_GGA(x1, x2, x3) = DIV_S_IN_GGA(x1, x2) U2_GGA(x1, x2, x3) = U2_GGA(x3) LSS_IN_GG(x1, x2) = LSS_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x3) U3_GGA(x1, x2, x3, x4) = U3_GGA(x2, x4) SUB_IN_GGA(x1, x2, x3) = SUB_IN_GGA(x1, x2) U6_GGA(x1, x2, x3, x4) = U6_GGA(x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: DIV_IN_GGA(X, s(Y), Z) -> U1_GGA(X, Y, Z, div_s_in_gga(X, Y, Z)) DIV_IN_GGA(X, s(Y), Z) -> DIV_S_IN_GGA(X, Y, Z) DIV_S_IN_GGA(s(X), Y, 0) -> U2_GGA(X, Y, lss_in_gg(X, Y)) DIV_S_IN_GGA(s(X), Y, 0) -> LSS_IN_GG(X, Y) LSS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, lss_in_gg(X, Y)) LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y) DIV_S_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, sub_in_gga(X, Y, R)) DIV_S_IN_GGA(s(X), Y, s(Z)) -> SUB_IN_GGA(X, Y, R) SUB_IN_GGA(s(X), s(Y), Z) -> U6_GGA(X, Y, Z, sub_in_gga(X, Y, Z)) SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z) U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_GGA(X, Y, Z, div_s_in_gga(R, Y, Z)) U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> DIV_S_IN_GGA(R, Y, Z) The TRS R consists of the following rules: div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z)) div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0) div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y)) lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y)) lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y)) U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y)) U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0) div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R)) sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z)) sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X) U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z) U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z)) U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z)) U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x4) div_s_in_gga(x1, x2, x3) = div_s_in_gga(x1, x2) 0 = 0 div_s_out_gga(x1, x2, x3) = div_s_out_gga(x3) U2_gga(x1, x2, x3) = U2_gga(x3) lss_in_gg(x1, x2) = lss_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) lss_out_gg(x1, x2) = lss_out_gg U3_gga(x1, x2, x3, x4) = U3_gga(x2, x4) sub_in_gga(x1, x2, x3) = sub_in_gga(x1, x2) U6_gga(x1, x2, x3, x4) = U6_gga(x4) sub_out_gga(x1, x2, x3) = sub_out_gga(x3) U4_gga(x1, x2, x3, x4) = U4_gga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) DIV_S_IN_GGA(x1, x2, x3) = DIV_S_IN_GGA(x1, x2) U2_GGA(x1, x2, x3) = U2_GGA(x3) LSS_IN_GG(x1, x2) = LSS_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x3) U3_GGA(x1, x2, x3, x4) = U3_GGA(x2, x4) SUB_IN_GGA(x1, x2, x3) = SUB_IN_GGA(x1, x2) U6_GGA(x1, x2, x3, x4) = U6_GGA(x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z) The TRS R consists of the following rules: div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z)) div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0) div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y)) lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y)) lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y)) U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y)) U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0) div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R)) sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z)) sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X) U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z) U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z)) U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z)) U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x4) div_s_in_gga(x1, x2, x3) = div_s_in_gga(x1, x2) 0 = 0 div_s_out_gga(x1, x2, x3) = div_s_out_gga(x3) U2_gga(x1, x2, x3) = U2_gga(x3) lss_in_gg(x1, x2) = lss_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) lss_out_gg(x1, x2) = lss_out_gg U3_gga(x1, x2, x3, x4) = U3_gga(x2, x4) sub_in_gga(x1, x2, x3) = sub_in_gga(x1, x2) U6_gga(x1, x2, x3, x4) = U6_gga(x4) sub_out_gga(x1, x2, x3) = sub_out_gga(x3) U4_gga(x1, x2, x3, x4) = U4_gga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) SUB_IN_GGA(x1, x2, x3) = SUB_IN_GGA(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: SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) SUB_IN_GGA(x1, x2, x3) = SUB_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) 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: SUB_IN_GGA(s(X), s(Y)) -> SUB_IN_GGA(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: *SUB_IN_GGA(s(X), s(Y)) -> SUB_IN_GGA(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: LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y) The TRS R consists of the following rules: div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z)) div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0) div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y)) lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y)) lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y)) U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y)) U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0) div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R)) sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z)) sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X) U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z) U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z)) U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z)) U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x4) div_s_in_gga(x1, x2, x3) = div_s_in_gga(x1, x2) 0 = 0 div_s_out_gga(x1, x2, x3) = div_s_out_gga(x3) U2_gga(x1, x2, x3) = U2_gga(x3) lss_in_gg(x1, x2) = lss_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) lss_out_gg(x1, x2) = lss_out_gg U3_gga(x1, x2, x3, x4) = U3_gga(x2, x4) sub_in_gga(x1, x2, x3) = sub_in_gga(x1, x2) U6_gga(x1, x2, x3, x4) = U6_gga(x4) sub_out_gga(x1, x2, x3) = sub_out_gga(x3) U4_gga(x1, x2, x3, x4) = U4_gga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) LSS_IN_GG(x1, x2) = LSS_IN_GG(x1, x2) 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: LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (EQUIVALENT) 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: LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y) R is empty. Q is empty. 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: *LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: DIV_S_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, sub_in_gga(X, Y, R)) U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> DIV_S_IN_GGA(R, Y, Z) The TRS R consists of the following rules: div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z)) div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0) div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y)) lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y)) lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y)) U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y)) U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0) div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R)) sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z)) sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X) U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z) U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z)) U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z)) U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x4) div_s_in_gga(x1, x2, x3) = div_s_in_gga(x1, x2) 0 = 0 div_s_out_gga(x1, x2, x3) = div_s_out_gga(x3) U2_gga(x1, x2, x3) = U2_gga(x3) lss_in_gg(x1, x2) = lss_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) lss_out_gg(x1, x2) = lss_out_gg U3_gga(x1, x2, x3, x4) = U3_gga(x2, x4) sub_in_gga(x1, x2, x3) = sub_in_gga(x1, x2) U6_gga(x1, x2, x3, x4) = U6_gga(x4) sub_out_gga(x1, x2, x3) = sub_out_gga(x3) U4_gga(x1, x2, x3, x4) = U4_gga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) DIV_S_IN_GGA(x1, x2, x3) = DIV_S_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x2, x4) 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: DIV_S_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, sub_in_gga(X, Y, R)) U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> DIV_S_IN_GGA(R, Y, Z) The TRS R consists of the following rules: sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z)) sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X) U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z) The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 sub_in_gga(x1, x2, x3) = sub_in_gga(x1, x2) U6_gga(x1, x2, x3, x4) = U6_gga(x4) sub_out_gga(x1, x2, x3) = sub_out_gga(x3) DIV_S_IN_GGA(x1, x2, x3) = DIV_S_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) 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: DIV_S_IN_GGA(s(X), Y) -> U3_GGA(Y, sub_in_gga(X, Y)) U3_GGA(Y, sub_out_gga(R)) -> DIV_S_IN_GGA(R, Y) The TRS R consists of the following rules: sub_in_gga(s(X), s(Y)) -> U6_gga(sub_in_gga(X, Y)) sub_in_gga(X, 0) -> sub_out_gga(X) U6_gga(sub_out_gga(Z)) -> sub_out_gga(Z) The set Q consists of the following terms: sub_in_gga(x0, x1) U6_gga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U3_GGA(Y, sub_out_gga(R)) -> DIV_S_IN_GGA(R, Y) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. DIV_S_IN_GGA(x1, x2) = x1 s(x1) = s(x1) U3_GGA(x1, x2) = x2 sub_in_gga(x1, x2) = sub_in_gga(x1) sub_out_gga(x1) = sub_out_gga(x1) U6_gga(x1) = x1 0 = 0 Recursive path order with status [RPO]. Quasi-Precedence: [s_1, sub_in_gga_1] > sub_out_gga_1 Status: s_1: multiset status sub_in_gga_1: multiset status sub_out_gga_1: multiset status 0: multiset status The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: sub_in_gga(s(X), s(Y)) -> U6_gga(sub_in_gga(X, Y)) sub_in_gga(X, 0) -> sub_out_gga(X) U6_gga(sub_out_gga(Z)) -> sub_out_gga(Z) ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: DIV_S_IN_GGA(s(X), Y) -> U3_GGA(Y, sub_in_gga(X, Y)) The TRS R consists of the following rules: sub_in_gga(s(X), s(Y)) -> U6_gga(sub_in_gga(X, Y)) sub_in_gga(X, 0) -> sub_out_gga(X) U6_gga(sub_out_gga(Z)) -> sub_out_gga(Z) The set Q consists of the following terms: sub_in_gga(x0, x1) U6_gga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (29) TRUE