/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern timeslist(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) UndefinedPredicateHandlerProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 0 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 0 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) PiDP (11) PiDPToQDPProof [SOUND, 15 ms] (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Clauses: timeslist(.(I, Is), Prod) :- timeslist(Is, I, Prod). timeslist([], Prod, Prod). timeslist(.(I, Is), Temp, Prod) :- ','(is(Temp1, *(Temp, I)), timeslist(Is, Temp1, Prod)). Query: timeslist(g,a) ---------------------------------------- (1) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: timeslist(.(I, Is), Prod) :- timeslist(Is, I, Prod). timeslist([], Prod, Prod). timeslist(.(I, Is), Temp, Prod) :- ','(is(Temp1, *(Temp, I)), timeslist(Is, Temp1, Prod)). is(X0, X1). Query: timeslist(g,a) ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: timeslist_in_2: (b,f) timeslist_in_3: (b,b,f) (b,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: timeslist_in_ga(.(I, Is), Prod) -> U1_ga(I, Is, Prod, timeslist_in_gga(Is, I, Prod)) timeslist_in_gga([], Prod, Prod) -> timeslist_out_gga([], Prod, Prod) timeslist_in_gga(.(I, Is), Temp, Prod) -> U2_gga(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gga(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) timeslist_in_gaa([], Prod, Prod) -> timeslist_out_gaa([], Prod, Prod) timeslist_in_gaa(.(I, Is), Temp, Prod) -> U2_gaa(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) U2_gaa(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gaa(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U3_gaa(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gaa(.(I, Is), Temp, Prod) U3_gga(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gga(.(I, Is), Temp, Prod) U1_ga(I, Is, Prod, timeslist_out_gga(Is, I, Prod)) -> timeslist_out_ga(.(I, Is), Prod) The argument filtering Pi contains the following mapping: timeslist_in_ga(x1, x2) = timeslist_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) timeslist_in_gga(x1, x2, x3) = timeslist_in_gga(x1, x2) [] = [] timeslist_out_gga(x1, x2, x3) = timeslist_out_gga U2_gga(x1, x2, x3, x4, x5) = U2_gga(x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag *(x1, x2) = *(x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) timeslist_in_gaa(x1, x2, x3) = timeslist_in_gaa(x1) timeslist_out_gaa(x1, x2, x3) = timeslist_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) timeslist_out_ga(x1, x2) = timeslist_out_ga Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: timeslist_in_ga(.(I, Is), Prod) -> U1_ga(I, Is, Prod, timeslist_in_gga(Is, I, Prod)) timeslist_in_gga([], Prod, Prod) -> timeslist_out_gga([], Prod, Prod) timeslist_in_gga(.(I, Is), Temp, Prod) -> U2_gga(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gga(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) timeslist_in_gaa([], Prod, Prod) -> timeslist_out_gaa([], Prod, Prod) timeslist_in_gaa(.(I, Is), Temp, Prod) -> U2_gaa(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) U2_gaa(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gaa(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U3_gaa(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gaa(.(I, Is), Temp, Prod) U3_gga(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gga(.(I, Is), Temp, Prod) U1_ga(I, Is, Prod, timeslist_out_gga(Is, I, Prod)) -> timeslist_out_ga(.(I, Is), Prod) The argument filtering Pi contains the following mapping: timeslist_in_ga(x1, x2) = timeslist_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) timeslist_in_gga(x1, x2, x3) = timeslist_in_gga(x1, x2) [] = [] timeslist_out_gga(x1, x2, x3) = timeslist_out_gga U2_gga(x1, x2, x3, x4, x5) = U2_gga(x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag *(x1, x2) = *(x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) timeslist_in_gaa(x1, x2, x3) = timeslist_in_gaa(x1) timeslist_out_gaa(x1, x2, x3) = timeslist_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) timeslist_out_ga(x1, x2) = timeslist_out_ga ---------------------------------------- (5) 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: TIMESLIST_IN_GA(.(I, Is), Prod) -> U1_GA(I, Is, Prod, timeslist_in_gga(Is, I, Prod)) TIMESLIST_IN_GA(.(I, Is), Prod) -> TIMESLIST_IN_GGA(Is, I, Prod) TIMESLIST_IN_GGA(.(I, Is), Temp, Prod) -> U2_GGA(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) TIMESLIST_IN_GGA(.(I, Is), Temp, Prod) -> IS_IN_AG(Temp1, *(Temp, I)) U2_GGA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_GGA(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U2_GGA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> TIMESLIST_IN_GAA(Is, Temp1, Prod) TIMESLIST_IN_GAA(.(I, Is), Temp, Prod) -> U2_GAA(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) TIMESLIST_IN_GAA(.(I, Is), Temp, Prod) -> IS_IN_AG(Temp1, *(Temp, I)) U2_GAA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_GAA(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U2_GAA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> TIMESLIST_IN_GAA(Is, Temp1, Prod) The TRS R consists of the following rules: timeslist_in_ga(.(I, Is), Prod) -> U1_ga(I, Is, Prod, timeslist_in_gga(Is, I, Prod)) timeslist_in_gga([], Prod, Prod) -> timeslist_out_gga([], Prod, Prod) timeslist_in_gga(.(I, Is), Temp, Prod) -> U2_gga(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gga(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) timeslist_in_gaa([], Prod, Prod) -> timeslist_out_gaa([], Prod, Prod) timeslist_in_gaa(.(I, Is), Temp, Prod) -> U2_gaa(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) U2_gaa(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gaa(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U3_gaa(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gaa(.(I, Is), Temp, Prod) U3_gga(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gga(.(I, Is), Temp, Prod) U1_ga(I, Is, Prod, timeslist_out_gga(Is, I, Prod)) -> timeslist_out_ga(.(I, Is), Prod) The argument filtering Pi contains the following mapping: timeslist_in_ga(x1, x2) = timeslist_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) timeslist_in_gga(x1, x2, x3) = timeslist_in_gga(x1, x2) [] = [] timeslist_out_gga(x1, x2, x3) = timeslist_out_gga U2_gga(x1, x2, x3, x4, x5) = U2_gga(x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag *(x1, x2) = *(x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) timeslist_in_gaa(x1, x2, x3) = timeslist_in_gaa(x1) timeslist_out_gaa(x1, x2, x3) = timeslist_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) timeslist_out_ga(x1, x2) = timeslist_out_ga TIMESLIST_IN_GA(x1, x2) = TIMESLIST_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) TIMESLIST_IN_GGA(x1, x2, x3) = TIMESLIST_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x2, x5) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x5) TIMESLIST_IN_GAA(x1, x2, x3) = TIMESLIST_IN_GAA(x1) U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x2, x5) U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMESLIST_IN_GA(.(I, Is), Prod) -> U1_GA(I, Is, Prod, timeslist_in_gga(Is, I, Prod)) TIMESLIST_IN_GA(.(I, Is), Prod) -> TIMESLIST_IN_GGA(Is, I, Prod) TIMESLIST_IN_GGA(.(I, Is), Temp, Prod) -> U2_GGA(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) TIMESLIST_IN_GGA(.(I, Is), Temp, Prod) -> IS_IN_AG(Temp1, *(Temp, I)) U2_GGA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_GGA(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U2_GGA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> TIMESLIST_IN_GAA(Is, Temp1, Prod) TIMESLIST_IN_GAA(.(I, Is), Temp, Prod) -> U2_GAA(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) TIMESLIST_IN_GAA(.(I, Is), Temp, Prod) -> IS_IN_AG(Temp1, *(Temp, I)) U2_GAA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_GAA(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U2_GAA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> TIMESLIST_IN_GAA(Is, Temp1, Prod) The TRS R consists of the following rules: timeslist_in_ga(.(I, Is), Prod) -> U1_ga(I, Is, Prod, timeslist_in_gga(Is, I, Prod)) timeslist_in_gga([], Prod, Prod) -> timeslist_out_gga([], Prod, Prod) timeslist_in_gga(.(I, Is), Temp, Prod) -> U2_gga(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gga(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) timeslist_in_gaa([], Prod, Prod) -> timeslist_out_gaa([], Prod, Prod) timeslist_in_gaa(.(I, Is), Temp, Prod) -> U2_gaa(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) U2_gaa(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gaa(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U3_gaa(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gaa(.(I, Is), Temp, Prod) U3_gga(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gga(.(I, Is), Temp, Prod) U1_ga(I, Is, Prod, timeslist_out_gga(Is, I, Prod)) -> timeslist_out_ga(.(I, Is), Prod) The argument filtering Pi contains the following mapping: timeslist_in_ga(x1, x2) = timeslist_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) timeslist_in_gga(x1, x2, x3) = timeslist_in_gga(x1, x2) [] = [] timeslist_out_gga(x1, x2, x3) = timeslist_out_gga U2_gga(x1, x2, x3, x4, x5) = U2_gga(x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag *(x1, x2) = *(x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) timeslist_in_gaa(x1, x2, x3) = timeslist_in_gaa(x1) timeslist_out_gaa(x1, x2, x3) = timeslist_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) timeslist_out_ga(x1, x2) = timeslist_out_ga TIMESLIST_IN_GA(x1, x2) = TIMESLIST_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) TIMESLIST_IN_GGA(x1, x2, x3) = TIMESLIST_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x2, x5) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x5) TIMESLIST_IN_GAA(x1, x2, x3) = TIMESLIST_IN_GAA(x1) U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x2, x5) U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 8 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_GAA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> TIMESLIST_IN_GAA(Is, Temp1, Prod) TIMESLIST_IN_GAA(.(I, Is), Temp, Prod) -> U2_GAA(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) The TRS R consists of the following rules: timeslist_in_ga(.(I, Is), Prod) -> U1_ga(I, Is, Prod, timeslist_in_gga(Is, I, Prod)) timeslist_in_gga([], Prod, Prod) -> timeslist_out_gga([], Prod, Prod) timeslist_in_gga(.(I, Is), Temp, Prod) -> U2_gga(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gga(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) timeslist_in_gaa([], Prod, Prod) -> timeslist_out_gaa([], Prod, Prod) timeslist_in_gaa(.(I, Is), Temp, Prod) -> U2_gaa(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) U2_gaa(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> U3_gaa(I, Is, Temp, Prod, timeslist_in_gaa(Is, Temp1, Prod)) U3_gaa(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gaa(.(I, Is), Temp, Prod) U3_gga(I, Is, Temp, Prod, timeslist_out_gaa(Is, Temp1, Prod)) -> timeslist_out_gga(.(I, Is), Temp, Prod) U1_ga(I, Is, Prod, timeslist_out_gga(Is, I, Prod)) -> timeslist_out_ga(.(I, Is), Prod) The argument filtering Pi contains the following mapping: timeslist_in_ga(x1, x2) = timeslist_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) timeslist_in_gga(x1, x2, x3) = timeslist_in_gga(x1, x2) [] = [] timeslist_out_gga(x1, x2, x3) = timeslist_out_gga U2_gga(x1, x2, x3, x4, x5) = U2_gga(x2, x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag *(x1, x2) = *(x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) timeslist_in_gaa(x1, x2, x3) = timeslist_in_gaa(x1) timeslist_out_gaa(x1, x2, x3) = timeslist_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) timeslist_out_ga(x1, x2) = timeslist_out_ga TIMESLIST_IN_GAA(x1, x2, x3) = TIMESLIST_IN_GAA(x1) U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (10) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_GAA(I, Is, Temp, Prod, is_out_ag(Temp1, *(Temp, I))) -> TIMESLIST_IN_GAA(Is, Temp1, Prod) TIMESLIST_IN_GAA(.(I, Is), Temp, Prod) -> U2_GAA(I, Is, Temp, Prod, is_in_ag(Temp1, *(Temp, I))) The TRS R consists of the following rules: is_in_ag(X0, X1) -> is_out_ag(X0, X1) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag *(x1, x2) = *(x2) TIMESLIST_IN_GAA(x1, x2, x3) = TIMESLIST_IN_GAA(x1) U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GAA(Is, is_out_ag) -> TIMESLIST_IN_GAA(Is) TIMESLIST_IN_GAA(.(I, Is)) -> U2_GAA(Is, is_in_ag(*(I))) The TRS R consists of the following rules: is_in_ag(X1) -> is_out_ag The set Q consists of the following terms: is_in_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) 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: *TIMESLIST_IN_GAA(.(I, Is)) -> U2_GAA(Is, is_in_ag(*(I))) The graph contains the following edges 1 > 1 *U2_GAA(Is, is_out_ag) -> TIMESLIST_IN_GAA(Is) The graph contains the following edges 1 >= 1 ---------------------------------------- (14) YES