/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Outermost Termination of the given OTRS could be proven: (0) OTRS (1) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 59 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 30 ms] (6) QTRS (7) AAECC Innermost [EQUIVALENT, 0 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 0 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) UsableRulesProof [EQUIVALENT, 0 ms] (15) QDP (16) QReductionProof [EQUIVALENT, 0 ms] (17) QDP (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) QDP (23) QReductionProof [EQUIVALENT, 0 ms] (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) DependencyGraphProof [EQUIVALENT, 0 ms] (28) QDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) QDP (31) TransformationProof [EQUIVALENT, 0 ms] (32) QDP (33) TransformationProof [EQUIVALENT, 0 ms] (34) QDP (35) UsableRulesProof [EQUIVALENT, 0 ms] (36) QDP (37) TransformationProof [EQUIVALENT, 0 ms] (38) QDP (39) DependencyGraphProof [EQUIVALENT, 0 ms] (40) QDP (41) TransformationProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 0 ms] (44) QDP (45) TransformationProof [EQUIVALENT, 0 ms] (46) QDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) QDP (49) TransformationProof [EQUIVALENT, 0 ms] (50) QDP (51) DependencyGraphProof [EQUIVALENT, 0 ms] (52) QDP (53) TransformationProof [EQUIVALENT, 0 ms] (54) QDP (55) TransformationProof [EQUIVALENT, 0 ms] (56) QDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) QDP (59) MRRProof [EQUIVALENT, 15 ms] (60) QDP (61) PisEmptyProof [EQUIVALENT, 0 ms] (62) YES ---------------------------------------- (0) Obligation: Term rewrite system R: The TRS R consists of the following rules: f(x, y) -> g(f(y, x)) g(g(g(f(x, y)))) -> x Outermost Strategy. ---------------------------------------- (1) Raffelsieper-Zantema-Transformation (SOUND) We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) down(g(g(g(f(x, y))))) -> up(x) top(up(x)) -> top(down(x)) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(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. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(block(x_1)) = 2 + 2*x_1 POL(down(x_1)) = 2*x_1 POL(f(x_1, x_2)) = 2*x_1 + 2*x_2 POL(f_flat(x_1, x_2)) = 2*x_1 + 2*x_2 POL(fresh_constant) = 0 POL(g(x_1)) = x_1 POL(g_flat(x_1)) = x_1 POL(top(x_1)) = 2*x_1 POL(up(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 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)) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) down(g(g(g(f(x, y))))) -> up(x) top(up(x)) -> top(down(x)) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(down(x_1)) = 2*x_1 POL(f(x_1, x_2)) = 1 + x_1 + x_2 POL(fresh_constant) = 0 POL(g(x_1)) = x_1 POL(g_flat(x_1)) = x_1 POL(top(x_1)) = 2*x_1 POL(up(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: down(g(g(g(f(x, y))))) -> up(x) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) top(up(x)) -> top(down(x)) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. ---------------------------------------- (7) AAECC Innermost (EQUIVALENT) We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) down(f(x, y)) -> up(g(f(y, x))) The TRS R 2 is top(up(x)) -> top(down(x)) The signature Sigma is {top_1} ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) top(up(x)) -> top(down(x)) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(f(x0, x1)) top(up(x0)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) ---------------------------------------- (9) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) TOP(up(x)) -> DOWN(x) DOWN(g(f(y4, y5))) -> G_FLAT(down(f(y4, y5))) DOWN(g(f(y4, y5))) -> DOWN(f(y4, y5)) DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) DOWN(g(fresh_constant)) -> DOWN(fresh_constant) DOWN(g(g(f(y8, y9)))) -> G_FLAT(down(g(f(y8, y9)))) DOWN(g(g(f(y8, y9)))) -> DOWN(g(f(y8, y9))) DOWN(g(g(fresh_constant))) -> G_FLAT(down(g(fresh_constant))) DOWN(g(g(fresh_constant))) -> DOWN(g(fresh_constant)) DOWN(g(g(g(g(y14))))) -> G_FLAT(down(g(g(g(y14))))) DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) DOWN(g(g(g(fresh_constant)))) -> G_FLAT(down(g(g(fresh_constant)))) DOWN(g(g(g(fresh_constant)))) -> DOWN(g(g(fresh_constant))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) top(up(x)) -> top(down(x)) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(f(x0, x1)) top(up(x0)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 12 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) top(up(x)) -> top(down(x)) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(f(x0, x1)) top(up(x0)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) R is empty. The set Q consists of the following terms: down(f(x0, x1)) top(up(x0)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) 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]. down(f(x0, x1)) top(up(x0)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) 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(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) The graph contains the following edges 1 > 1 ---------------------------------------- (19) YES ---------------------------------------- (20) 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, y)) -> up(g(f(y, x))) top(up(x)) -> top(down(x)) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(f(x0, x1)) top(up(x0)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(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(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(f(x0, x1)) top(up(x0)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(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(up(x0)) ---------------------------------------- (24) 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, y)) -> up(g(f(y, x))) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) 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, x1))) -> TOP(up(g(f(x1, x0)))),TOP(up(f(x0, x1))) -> TOP(up(g(f(x1, x0))))) (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(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) (TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))),TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1)))))) (TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))),TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))) (TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))),TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0))))))) (TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))),TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant)))))) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(x0, x1))) -> TOP(up(g(f(x1, x0)))) TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (28) 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(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) 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. ---------------------------------------- (30) 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(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) The TRS R consists of the following rules: down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(f(x, y)) -> up(g(f(y, x))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))),TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0)))))) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) The TRS R consists of the following rules: down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(f(x, y)) -> up(g(f(y, x))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))),TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1)))))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) The TRS R consists of the following rules: down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) down(f(x, y)) -> up(g(f(y, x))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) 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. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) g_flat(up(x_1)) -> up(g(x_1)) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))),TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant))))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) g_flat(up(x_1)) -> up(g(x_1)) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) g_flat(up(x_1)) -> up(g(x_1)) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))),TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant)))))) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) g_flat(up(x_1)) -> up(g(x_1)) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))),TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0)))))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) g_flat(up(x_1)) -> up(g(x_1)) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))),TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0))))))) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) The TRS R consists of the following rules: down(f(x, y)) -> up(g(f(y, x))) g_flat(up(x_1)) -> up(g(x_1)) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) 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. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(g_flat(down(fresh_constant))))),TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(g_flat(down(fresh_constant)))))) ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(g_flat(down(fresh_constant))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(up(g(g(f(x1, x0)))))),TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(up(g(g(f(x1, x0))))))) ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(up(g(g(f(x1, x0)))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(up(g(g(f(x1, x0)))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(f(x0, x1))))) -> TOP(up(g(g(g(f(x1, x0)))))),TOP(up(g(g(f(x0, x1))))) -> TOP(up(g(g(g(f(x1, x0))))))) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) TOP(up(g(g(f(x0, x1))))) -> TOP(up(g(g(g(f(x1, x0)))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) 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 dependency pairs: TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) Strictly oriented rules of the TRS R: g_flat(up(x_1)) -> up(g(x_1)) Used ordering: Polynomial interpretation [POLO]: POL(TOP(x_1)) = 2*x_1 POL(down(x_1)) = x_1 POL(fresh_constant) = 0 POL(g(x_1)) = 2*x_1 POL(g_flat(x_1)) = 2*x_1 POL(up(x_1)) = 2 + x_1 ---------------------------------------- (60) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) The set Q consists of the following terms: down(f(x0, x1)) down(g(f(x0, x1))) down(g(fresh_constant)) down(g(g(f(x0, x1)))) down(g(g(fresh_constant))) down(g(g(g(g(x0))))) down(g(g(g(fresh_constant)))) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (62) YES