/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern my_length(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 4 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: my_length([], 0). my_length(.(X1, L), N) :- ','(my_length(L, N1), is(N, +(N1, 1))). Query: my_length(g,a) ---------------------------------------- (1) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: my_length([], 0). my_length(.(X1, L), N) :- ','(my_length(L, N1), is(N, +(N1, 1))). is(X0, X1). Query: my_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: my_length_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: my_length_in_ga([], 0) -> my_length_out_ga([], 0) my_length_in_ga(.(X1, L), N) -> U1_ga(X1, L, N, my_length_in_ga(L, N1)) U1_ga(X1, L, N, my_length_out_ga(L, N1)) -> U2_ga(X1, L, N, N1, is_in_ag(N, +(N1, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ga(X1, L, N, N1, is_out_ag(N, +(N1, 1))) -> my_length_out_ga(.(X1, L), N) The argument filtering Pi contains the following mapping: my_length_in_ga(x1, x2) = my_length_in_ga(x1) [] = [] my_length_out_ga(x1, x2) = my_length_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x2) 1 = 1 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: my_length_in_ga([], 0) -> my_length_out_ga([], 0) my_length_in_ga(.(X1, L), N) -> U1_ga(X1, L, N, my_length_in_ga(L, N1)) U1_ga(X1, L, N, my_length_out_ga(L, N1)) -> U2_ga(X1, L, N, N1, is_in_ag(N, +(N1, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ga(X1, L, N, N1, is_out_ag(N, +(N1, 1))) -> my_length_out_ga(.(X1, L), N) The argument filtering Pi contains the following mapping: my_length_in_ga(x1, x2) = my_length_in_ga(x1) [] = [] my_length_out_ga(x1, x2) = my_length_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x2) 1 = 1 ---------------------------------------- (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: MY_LENGTH_IN_GA(.(X1, L), N) -> U1_GA(X1, L, N, my_length_in_ga(L, N1)) MY_LENGTH_IN_GA(.(X1, L), N) -> MY_LENGTH_IN_GA(L, N1) U1_GA(X1, L, N, my_length_out_ga(L, N1)) -> U2_GA(X1, L, N, N1, is_in_ag(N, +(N1, 1))) U1_GA(X1, L, N, my_length_out_ga(L, N1)) -> IS_IN_AG(N, +(N1, 1)) The TRS R consists of the following rules: my_length_in_ga([], 0) -> my_length_out_ga([], 0) my_length_in_ga(.(X1, L), N) -> U1_ga(X1, L, N, my_length_in_ga(L, N1)) U1_ga(X1, L, N, my_length_out_ga(L, N1)) -> U2_ga(X1, L, N, N1, is_in_ag(N, +(N1, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ga(X1, L, N, N1, is_out_ag(N, +(N1, 1))) -> my_length_out_ga(.(X1, L), N) The argument filtering Pi contains the following mapping: my_length_in_ga(x1, x2) = my_length_in_ga(x1) [] = [] my_length_out_ga(x1, x2) = my_length_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x2) 1 = 1 MY_LENGTH_IN_GA(x1, x2) = MY_LENGTH_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5) IS_IN_AG(x1, x2) = IS_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: MY_LENGTH_IN_GA(.(X1, L), N) -> U1_GA(X1, L, N, my_length_in_ga(L, N1)) MY_LENGTH_IN_GA(.(X1, L), N) -> MY_LENGTH_IN_GA(L, N1) U1_GA(X1, L, N, my_length_out_ga(L, N1)) -> U2_GA(X1, L, N, N1, is_in_ag(N, +(N1, 1))) U1_GA(X1, L, N, my_length_out_ga(L, N1)) -> IS_IN_AG(N, +(N1, 1)) The TRS R consists of the following rules: my_length_in_ga([], 0) -> my_length_out_ga([], 0) my_length_in_ga(.(X1, L), N) -> U1_ga(X1, L, N, my_length_in_ga(L, N1)) U1_ga(X1, L, N, my_length_out_ga(L, N1)) -> U2_ga(X1, L, N, N1, is_in_ag(N, +(N1, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ga(X1, L, N, N1, is_out_ag(N, +(N1, 1))) -> my_length_out_ga(.(X1, L), N) The argument filtering Pi contains the following mapping: my_length_in_ga(x1, x2) = my_length_in_ga(x1) [] = [] my_length_out_ga(x1, x2) = my_length_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x2) 1 = 1 MY_LENGTH_IN_GA(x1, x2) = MY_LENGTH_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5) IS_IN_AG(x1, x2) = IS_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: MY_LENGTH_IN_GA(.(X1, L), N) -> MY_LENGTH_IN_GA(L, N1) The TRS R consists of the following rules: my_length_in_ga([], 0) -> my_length_out_ga([], 0) my_length_in_ga(.(X1, L), N) -> U1_ga(X1, L, N, my_length_in_ga(L, N1)) U1_ga(X1, L, N, my_length_out_ga(L, N1)) -> U2_ga(X1, L, N, N1, is_in_ag(N, +(N1, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U2_ga(X1, L, N, N1, is_out_ag(N, +(N1, 1))) -> my_length_out_ga(.(X1, L), N) The argument filtering Pi contains the following mapping: my_length_in_ga(x1, x2) = my_length_in_ga(x1) [] = [] my_length_out_ga(x1, x2) = my_length_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x2) 1 = 1 MY_LENGTH_IN_GA(x1, x2) = MY_LENGTH_IN_GA(x1) 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: MY_LENGTH_IN_GA(.(X1, L), N) -> MY_LENGTH_IN_GA(L, N1) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MY_LENGTH_IN_GA(x1, x2) = MY_LENGTH_IN_GA(x1) 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: MY_LENGTH_IN_GA(.(X1, L)) -> MY_LENGTH_IN_GA(L) R is empty. Q is empty. 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: *MY_LENGTH_IN_GA(.(X1, L)) -> MY_LENGTH_IN_GA(L) The graph contains the following edges 1 > 1 ---------------------------------------- (14) YES