/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern length(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 18 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, 0 ms] (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Clauses: length(Xs, L) :- length(Xs, 0, L). length([], L, L). length(.(X, Xs), L0, L) :- ','(is(L1, +(L0, 1)), length(Xs, L1, L)). Query: length(g,a) ---------------------------------------- (1) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: length(Xs, L) :- length(Xs, 0, L). length([], L, L). length(.(X, Xs), L0, L) :- ','(is(L1, +(L0, 1)), length(Xs, L1, L)). is(X0, X1). Query: length(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: length_in_2: (b,f) length_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: length_in_ga(Xs, L) -> U1_ga(Xs, L, length_in_gga(Xs, 0, L)) length_in_gga([], L, L) -> length_out_gga([], L, L) length_in_gga(.(X, Xs), L0, L) -> U2_gga(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gga(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) length_in_gaa([], L, L) -> length_out_gaa([], L, L) length_in_gaa(.(X, Xs), L0, L) -> U2_gaa(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) U2_gaa(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gaa(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U3_gaa(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gaa(.(X, Xs), L0, L) U3_gga(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gga(.(X, Xs), L0, L) U1_ga(Xs, L, length_out_gga(Xs, 0, L)) -> length_out_ga(Xs, L) The argument filtering Pi contains the following mapping: length_in_ga(x1, x2) = length_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) length_in_gga(x1, x2, x3) = length_in_gga(x1, x2) [] = [] length_out_gga(x1, x2, x3) = length_out_gga .(x1, x2) = .(x1, x2) 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) 1 = 1 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) length_in_gaa(x1, x2, x3) = length_in_gaa(x1) length_out_gaa(x1, x2, x3) = length_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) 0 = 0 length_out_ga(x1, x2) = length_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: length_in_ga(Xs, L) -> U1_ga(Xs, L, length_in_gga(Xs, 0, L)) length_in_gga([], L, L) -> length_out_gga([], L, L) length_in_gga(.(X, Xs), L0, L) -> U2_gga(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gga(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) length_in_gaa([], L, L) -> length_out_gaa([], L, L) length_in_gaa(.(X, Xs), L0, L) -> U2_gaa(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) U2_gaa(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gaa(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U3_gaa(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gaa(.(X, Xs), L0, L) U3_gga(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gga(.(X, Xs), L0, L) U1_ga(Xs, L, length_out_gga(Xs, 0, L)) -> length_out_ga(Xs, L) The argument filtering Pi contains the following mapping: length_in_ga(x1, x2) = length_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) length_in_gga(x1, x2, x3) = length_in_gga(x1, x2) [] = [] length_out_gga(x1, x2, x3) = length_out_gga .(x1, x2) = .(x1, x2) 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) 1 = 1 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) length_in_gaa(x1, x2, x3) = length_in_gaa(x1) length_out_gaa(x1, x2, x3) = length_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) 0 = 0 length_out_ga(x1, x2) = length_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: LENGTH_IN_GA(Xs, L) -> U1_GA(Xs, L, length_in_gga(Xs, 0, L)) LENGTH_IN_GA(Xs, L) -> LENGTH_IN_GGA(Xs, 0, L) LENGTH_IN_GGA(.(X, Xs), L0, L) -> U2_GGA(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) LENGTH_IN_GGA(.(X, Xs), L0, L) -> IS_IN_AG(L1, +(L0, 1)) U2_GGA(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_GGA(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U2_GGA(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> LENGTH_IN_GAA(Xs, L1, L) LENGTH_IN_GAA(.(X, Xs), L0, L) -> U2_GAA(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) LENGTH_IN_GAA(.(X, Xs), L0, L) -> IS_IN_AG(L1, +(L0, 1)) U2_GAA(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_GAA(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U2_GAA(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> LENGTH_IN_GAA(Xs, L1, L) The TRS R consists of the following rules: length_in_ga(Xs, L) -> U1_ga(Xs, L, length_in_gga(Xs, 0, L)) length_in_gga([], L, L) -> length_out_gga([], L, L) length_in_gga(.(X, Xs), L0, L) -> U2_gga(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gga(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) length_in_gaa([], L, L) -> length_out_gaa([], L, L) length_in_gaa(.(X, Xs), L0, L) -> U2_gaa(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) U2_gaa(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gaa(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U3_gaa(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gaa(.(X, Xs), L0, L) U3_gga(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gga(.(X, Xs), L0, L) U1_ga(Xs, L, length_out_gga(Xs, 0, L)) -> length_out_ga(Xs, L) The argument filtering Pi contains the following mapping: length_in_ga(x1, x2) = length_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) length_in_gga(x1, x2, x3) = length_in_gga(x1, x2) [] = [] length_out_gga(x1, x2, x3) = length_out_gga .(x1, x2) = .(x1, x2) 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) 1 = 1 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) length_in_gaa(x1, x2, x3) = length_in_gaa(x1) length_out_gaa(x1, x2, x3) = length_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) 0 = 0 length_out_ga(x1, x2) = length_out_ga LENGTH_IN_GA(x1, x2) = LENGTH_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) LENGTH_IN_GGA(x1, x2, x3) = LENGTH_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) LENGTH_IN_GAA(x1, x2, x3) = LENGTH_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: LENGTH_IN_GA(Xs, L) -> U1_GA(Xs, L, length_in_gga(Xs, 0, L)) LENGTH_IN_GA(Xs, L) -> LENGTH_IN_GGA(Xs, 0, L) LENGTH_IN_GGA(.(X, Xs), L0, L) -> U2_GGA(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) LENGTH_IN_GGA(.(X, Xs), L0, L) -> IS_IN_AG(L1, +(L0, 1)) U2_GGA(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_GGA(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U2_GGA(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> LENGTH_IN_GAA(Xs, L1, L) LENGTH_IN_GAA(.(X, Xs), L0, L) -> U2_GAA(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) LENGTH_IN_GAA(.(X, Xs), L0, L) -> IS_IN_AG(L1, +(L0, 1)) U2_GAA(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_GAA(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U2_GAA(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> LENGTH_IN_GAA(Xs, L1, L) The TRS R consists of the following rules: length_in_ga(Xs, L) -> U1_ga(Xs, L, length_in_gga(Xs, 0, L)) length_in_gga([], L, L) -> length_out_gga([], L, L) length_in_gga(.(X, Xs), L0, L) -> U2_gga(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gga(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) length_in_gaa([], L, L) -> length_out_gaa([], L, L) length_in_gaa(.(X, Xs), L0, L) -> U2_gaa(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) U2_gaa(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gaa(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U3_gaa(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gaa(.(X, Xs), L0, L) U3_gga(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gga(.(X, Xs), L0, L) U1_ga(Xs, L, length_out_gga(Xs, 0, L)) -> length_out_ga(Xs, L) The argument filtering Pi contains the following mapping: length_in_ga(x1, x2) = length_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) length_in_gga(x1, x2, x3) = length_in_gga(x1, x2) [] = [] length_out_gga(x1, x2, x3) = length_out_gga .(x1, x2) = .(x1, x2) 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) 1 = 1 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) length_in_gaa(x1, x2, x3) = length_in_gaa(x1) length_out_gaa(x1, x2, x3) = length_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) 0 = 0 length_out_ga(x1, x2) = length_out_ga LENGTH_IN_GA(x1, x2) = LENGTH_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) LENGTH_IN_GGA(x1, x2, x3) = LENGTH_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) LENGTH_IN_GAA(x1, x2, x3) = LENGTH_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(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> LENGTH_IN_GAA(Xs, L1, L) LENGTH_IN_GAA(.(X, Xs), L0, L) -> U2_GAA(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) The TRS R consists of the following rules: length_in_ga(Xs, L) -> U1_ga(Xs, L, length_in_gga(Xs, 0, L)) length_in_gga([], L, L) -> length_out_gga([], L, L) length_in_gga(.(X, Xs), L0, L) -> U2_gga(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_gga(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gga(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) length_in_gaa([], L, L) -> length_out_gaa([], L, L) length_in_gaa(.(X, Xs), L0, L) -> U2_gaa(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) U2_gaa(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> U3_gaa(X, Xs, L0, L, length_in_gaa(Xs, L1, L)) U3_gaa(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gaa(.(X, Xs), L0, L) U3_gga(X, Xs, L0, L, length_out_gaa(Xs, L1, L)) -> length_out_gga(.(X, Xs), L0, L) U1_ga(Xs, L, length_out_gga(Xs, 0, L)) -> length_out_ga(Xs, L) The argument filtering Pi contains the following mapping: length_in_ga(x1, x2) = length_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) length_in_gga(x1, x2, x3) = length_in_gga(x1, x2) [] = [] length_out_gga(x1, x2, x3) = length_out_gga .(x1, x2) = .(x1, x2) 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) 1 = 1 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5) length_in_gaa(x1, x2, x3) = length_in_gaa(x1) length_out_gaa(x1, x2, x3) = length_out_gaa U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x2, x5) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) 0 = 0 length_out_ga(x1, x2) = length_out_ga LENGTH_IN_GAA(x1, x2, x3) = LENGTH_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(X, Xs, L0, L, is_out_ag(L1, +(L0, 1))) -> LENGTH_IN_GAA(Xs, L1, L) LENGTH_IN_GAA(.(X, Xs), L0, L) -> U2_GAA(X, Xs, L0, L, is_in_ag(L1, +(L0, 1))) 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) 1 = 1 LENGTH_IN_GAA(x1, x2, x3) = LENGTH_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(Xs, is_out_ag) -> LENGTH_IN_GAA(Xs) LENGTH_IN_GAA(.(X, Xs)) -> U2_GAA(Xs, is_in_ag(+(1))) 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: *LENGTH_IN_GAA(.(X, Xs)) -> U2_GAA(Xs, is_in_ag(+(1))) The graph contains the following edges 1 > 1 *U2_GAA(Xs, is_out_ag) -> LENGTH_IN_GAA(Xs) The graph contains the following edges 1 >= 1 ---------------------------------------- (14) YES