/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Outermost Termination of the given OTRS could not be shown: (0) OTRS (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) QDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) QDP (10) QReductionProof [EQUIVALENT, 0 ms] (11) QDP (12) MRRProof [EQUIVALENT, 11 ms] (13) QDP (14) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (15) QDP (16) UsableRulesReductionPairsProof [EQUIVALENT, 2 ms] (17) QDP (18) DependencyGraphProof [EQUIVALENT, 0 ms] (19) TRUE (20) QDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) QDP (23) QReductionProof [EQUIVALENT, 0 ms] (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) UsableRulesProof [EQUIVALENT, 0 ms] (28) QDP (29) QReductionProof [EQUIVALENT, 0 ms] (30) QDP (31) Trivial-Transformation [SOUND, 0 ms] (32) QTRS (33) DependencyPairsProof [EQUIVALENT, 0 ms] (34) QDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) QDP (37) NonTerminationLoopProof [COMPLETE, 0 ms] (38) NO (39) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] (40) QTRS (41) DependencyPairsProof [EQUIVALENT, 0 ms] (42) QDP (43) DependencyGraphProof [EQUIVALENT, 0 ms] (44) AND (45) QDP (46) UsableRulesProof [EQUIVALENT, 0 ms] (47) QDP (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] (49) YES (50) QDP (51) TransformationProof [EQUIVALENT, 0 ms] (52) QDP (53) DependencyGraphProof [EQUIVALENT, 0 ms] (54) QDP (55) QDPOrderProof [EQUIVALENT, 20 ms] (56) QDP ---------------------------------------- (0) Obligation: Term rewrite system R: The TRS R consists of the following rules: f(x, c) -> g(f(x, x)) g(g(x)) -> c Outermost Strategy. ---------------------------------------- (1) Thiemann-SpecialC-Transformation (EQUIVALENT) We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) redex_g(g(x)) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) TOP(go_up(x)) -> REDUCE(x) REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) REDUCE(f(x_1, x_2)) -> REDEX_F(x_1, x_2) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) REDUCE(g(x_1)) -> REDEX_G(x_1) CHECK_F(redex_f(x_1, x_2)) -> IN_F_1(reduce(x_1), x_2) CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) CHECK_F(redex_f(x_1, x_2)) -> IN_F_2(x_1, reduce(x_2)) CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) redex_g(g(x)) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) redex_g(g(x)) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_g(g(x)) -> result_g(c) redex_f(x, c) -> result_f(g(f(x, x))) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) reduce(f(x0, x1)) reduce(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_g(g(x)) -> result_g(c) redex_f(x, c) -> result_f(g(f(x, x))) The set Q consists of the following terms: redex_f(x0, c) redex_g(g(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: redex_g(g(x)) -> result_g(c) Used ordering: Polynomial interpretation [POLO]: POL(CHECK_F(x_1)) = 2 + x_1 POL(CHECK_G(x_1)) = x_1 POL(REDUCE(x_1)) = 2 + 2*x_1 POL(c) = 0 POL(f(x_1, x_2)) = x_1 + x_2 POL(g(x_1)) = x_1 POL(redex_f(x_1, x_2)) = 2*x_1 + 2*x_2 POL(redex_g(x_1)) = 2 + 2*x_1 POL(result_f(x_1)) = x_1 POL(result_g(x_1)) = 1 + 2*x_1 ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_f(x, c) -> result_f(g(f(x, x))) The set Q consists of the following terms: redex_f(x0, c) redex_g(g(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: redex_f(x, c) -> result_f(g(f(x, x))) Used ordering: POLO with Polynomial interpretation [POLO]: POL(CHECK_F(x_1)) = x_1 POL(CHECK_G(x_1)) = 2*x_1 POL(REDUCE(x_1)) = 2*x_1 POL(c) = 0 POL(f(x_1, x_2)) = x_1 + x_2 POL(g(x_1)) = x_1 POL(redex_f(x_1, x_2)) = 2*x_1 + 2*x_2 POL(redex_g(x_1)) = x_1 POL(result_f(x_1)) = x_1 ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) R is empty. The set Q consists of the following terms: redex_f(x0, c) redex_g(g(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) No rules are removed from R. Used ordering: POLO with Polynomial interpretation [POLO]: POL(CHECK_F(x_1)) = x_1 POL(CHECK_G(x_1)) = 2*x_1 POL(REDUCE(x_1)) = 2*x_1 POL(f(x_1, x_2)) = 2*x_1 + 2*x_2 POL(g(x_1)) = 2*x_1 POL(redex_f(x_1, x_2)) = 2*x_1 + 2*x_2 POL(redex_g(x_1)) = x_1 ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) R is empty. The set Q consists of the following terms: redex_f(x0, c) redex_g(g(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (19) TRUE ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) redex_g(g(x)) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_g(g(x)) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_g(g(x)) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) The set Q consists of the following terms: reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: (TOP(go_up(f(x0, x1))) -> TOP(check_f(redex_f(x0, x1))),TOP(go_up(f(x0, x1))) -> TOP(check_f(redex_f(x0, x1)))) (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(f(x0, x1))) -> TOP(check_f(redex_f(x0, x1))) TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) The TRS R consists of the following rules: reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_g(g(x)) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) The set Q consists of the following terms: reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_g(g(x)) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) reduce(g(x_1)) -> check_g(redex_g(x_1)) redex_g(g(x)) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(x, c) -> result_f(g(f(x, x))) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) The set Q consists of the following terms: reduce(f(x0, x1)) reduce(g(x0)) redex_f(x0, c) redex_g(g(x0)) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0, x1)) check_g(redex_g(x0)) in_f_1(go_up(x0), x1) in_f_2(x0, go_up(x1)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) Trivial-Transformation (SOUND) We applied the Trivial transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (32) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f(x, c) -> g(f(x, x)) g(g(x)) -> c Q is empty. ---------------------------------------- (33) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: F(x, c) -> G(f(x, x)) F(x, c) -> F(x, x) The TRS R consists of the following rules: f(x, c) -> g(f(x, x)) g(g(x)) -> c Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: F(x, c) -> F(x, x) The TRS R consists of the following rules: f(x, c) -> g(f(x, x)) g(g(x)) -> c Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = F(x, c) evaluates to t =F(x, x) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [x / c] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from F(c, c) to F(c, c). ---------------------------------------- (38) NO ---------------------------------------- (39) Raffelsieper-Zantema-Transformation (SOUND) We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (40) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(f(x, c)) -> up(g(f(x, x))) down(g(g(x))) -> up(c) top(up(x)) -> top(down(x)) down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. ---------------------------------------- (41) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) TOP(up(x)) -> DOWN(x) DOWN(f(y12, f(y13, y14))) -> F_FLAT(down(y12), block(f(y13, y14))) DOWN(f(y12, f(y13, y14))) -> DOWN(y12) DOWN(f(y12, f(y13, y14))) -> F_FLAT(block(y12), down(f(y13, y14))) DOWN(f(y12, f(y13, y14))) -> DOWN(f(y13, y14)) DOWN(f(y30, g(y31))) -> F_FLAT(down(y30), block(g(y31))) DOWN(f(y30, g(y31))) -> DOWN(y30) DOWN(f(y30, g(y31))) -> F_FLAT(block(y30), down(g(y31))) DOWN(f(y30, g(y31))) -> DOWN(g(y31)) DOWN(f(y39, fresh_constant)) -> F_FLAT(down(y39), block(fresh_constant)) DOWN(f(y39, fresh_constant)) -> DOWN(y39) DOWN(f(y39, fresh_constant)) -> F_FLAT(block(y39), down(fresh_constant)) DOWN(f(y39, fresh_constant)) -> DOWN(fresh_constant) DOWN(g(f(y44, y45))) -> G_FLAT(down(f(y44, y45))) DOWN(g(f(y44, y45))) -> DOWN(f(y44, y45)) DOWN(g(c)) -> G_FLAT(down(c)) DOWN(g(c)) -> DOWN(c) DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) DOWN(g(fresh_constant)) -> DOWN(fresh_constant) The TRS R consists of the following rules: down(f(x, c)) -> up(g(f(x, x))) down(g(g(x))) -> up(c) top(up(x)) -> top(down(x)) down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 13 less nodes. ---------------------------------------- (44) Complex Obligation (AND) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(f(y12, f(y13, y14))) -> DOWN(f(y13, y14)) DOWN(f(y12, f(y13, y14))) -> DOWN(y12) DOWN(f(y30, g(y31))) -> DOWN(y30) DOWN(f(y30, g(y31))) -> DOWN(g(y31)) DOWN(g(f(y44, y45))) -> DOWN(f(y44, y45)) DOWN(f(y39, fresh_constant)) -> DOWN(y39) The TRS R consists of the following rules: down(f(x, c)) -> up(g(f(x, x))) down(g(g(x))) -> up(c) top(up(x)) -> top(down(x)) down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(f(y12, f(y13, y14))) -> DOWN(f(y13, y14)) DOWN(f(y12, f(y13, y14))) -> DOWN(y12) DOWN(f(y30, g(y31))) -> DOWN(y30) DOWN(f(y30, g(y31))) -> DOWN(g(y31)) DOWN(g(f(y44, y45))) -> DOWN(f(y44, y45)) DOWN(f(y39, fresh_constant)) -> DOWN(y39) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) 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: *DOWN(f(y30, g(y31))) -> DOWN(g(y31)) The graph contains the following edges 1 > 1 *DOWN(g(f(y44, y45))) -> DOWN(f(y44, y45)) The graph contains the following edges 1 > 1 *DOWN(f(y12, f(y13, y14))) -> DOWN(f(y13, y14)) The graph contains the following edges 1 > 1 *DOWN(f(y12, f(y13, y14))) -> DOWN(y12) The graph contains the following edges 1 > 1 *DOWN(f(y30, g(y31))) -> DOWN(y30) The graph contains the following edges 1 > 1 *DOWN(f(y39, fresh_constant)) -> DOWN(y39) The graph contains the following edges 1 > 1 ---------------------------------------- (49) YES ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(f(x, c)) -> up(g(f(x, x))) down(g(g(x))) -> up(c) top(up(x)) -> top(down(x)) down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0)))),TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0))))) (TOP(up(g(g(x0)))) -> TOP(up(c)),TOP(up(g(g(x0)))) -> TOP(up(c))) (TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(down(x0), block(f(x1, x2)))),TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(down(x0), block(f(x1, x2))))) (TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(block(x0), down(f(x1, x2)))),TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(block(x0), down(f(x1, x2))))) (TOP(up(f(x0, g(x1)))) -> TOP(f_flat(down(x0), block(g(x1)))),TOP(up(f(x0, g(x1)))) -> TOP(f_flat(down(x0), block(g(x1))))) (TOP(up(f(x0, g(x1)))) -> TOP(f_flat(block(x0), down(g(x1)))),TOP(up(f(x0, g(x1)))) -> TOP(f_flat(block(x0), down(g(x1))))) (TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(down(x0), block(fresh_constant))),TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(down(x0), block(fresh_constant)))) (TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(block(x0), down(fresh_constant))),TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(block(x0), down(fresh_constant)))) (TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))),TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1))))) (TOP(up(g(c))) -> TOP(g_flat(down(c))),TOP(up(g(c))) -> TOP(g_flat(down(c)))) (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0)))) TOP(up(g(g(x0)))) -> TOP(up(c)) TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(down(x0), block(f(x1, x2)))) TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(block(x0), down(f(x1, x2)))) TOP(up(f(x0, g(x1)))) -> TOP(f_flat(down(x0), block(g(x1)))) TOP(up(f(x0, g(x1)))) -> TOP(f_flat(block(x0), down(g(x1)))) TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(down(x0), block(fresh_constant))) TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(block(x0), down(fresh_constant))) TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) TOP(up(g(c))) -> TOP(g_flat(down(c))) TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) The TRS R consists of the following rules: down(f(x, c)) -> up(g(f(x, x))) down(g(g(x))) -> up(c) top(up(x)) -> top(down(x)) down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0)))) TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(down(x0), block(f(x1, x2)))) TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(block(x0), down(f(x1, x2)))) TOP(up(f(x0, g(x1)))) -> TOP(f_flat(down(x0), block(g(x1)))) TOP(up(f(x0, g(x1)))) -> TOP(f_flat(block(x0), down(g(x1)))) TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(down(x0), block(fresh_constant))) The TRS R consists of the following rules: down(f(x, c)) -> up(g(f(x, x))) down(g(g(x))) -> up(c) top(up(x)) -> top(down(x)) down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP(x_1)) = x_1 POL(block(x_1)) = 0 POL(c) = 0 POL(down(x_1)) = 0 POL(f(x_1, x_2)) = 1 POL(f_flat(x_1, x_2)) = 1 POL(fresh_constant) = 0 POL(g(x_1)) = 0 POL(g_flat(x_1)) = 0 POL(up(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(down(x0), block(f(x1, x2)))) TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(block(x0), down(f(x1, x2)))) TOP(up(f(x0, g(x1)))) -> TOP(f_flat(down(x0), block(g(x1)))) TOP(up(f(x0, g(x1)))) -> TOP(f_flat(block(x0), down(g(x1)))) TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(down(x0), block(fresh_constant))) The TRS R consists of the following rules: down(f(x, c)) -> up(g(f(x, x))) down(g(g(x))) -> up(c) top(up(x)) -> top(down(x)) down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. We have to consider all minimal (P,Q,R)-chains.