5.88/2.26 YES 5.88/2.27 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 5.88/2.27 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.88/2.27 5.88/2.27 5.88/2.27 Outermost Termination of the given OTRS could be proven: 5.88/2.27 5.88/2.27 (0) OTRS 5.88/2.27 (1) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 5.88/2.27 (2) QTRS 5.88/2.27 (3) QTRSRRRProof [EQUIVALENT, 84 ms] 5.88/2.27 (4) QTRS 5.88/2.27 (5) QTRSRRRProof [EQUIVALENT, 32 ms] 5.88/2.27 (6) QTRS 5.88/2.27 (7) AAECC Innermost [EQUIVALENT, 0 ms] 5.88/2.27 (8) QTRS 5.88/2.27 (9) DependencyPairsProof [EQUIVALENT, 0 ms] 5.88/2.27 (10) QDP 5.88/2.27 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 5.88/2.27 (12) AND 5.88/2.27 (13) QDP 5.88/2.27 (14) UsableRulesProof [EQUIVALENT, 0 ms] 5.88/2.27 (15) QDP 5.88/2.27 (16) QReductionProof [EQUIVALENT, 0 ms] 5.88/2.27 (17) QDP 5.88/2.27 (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.88/2.27 (19) YES 5.88/2.27 (20) QDP 5.88/2.27 (21) UsableRulesProof [EQUIVALENT, 0 ms] 5.88/2.27 (22) QDP 5.88/2.27 (23) QReductionProof [EQUIVALENT, 0 ms] 5.88/2.27 (24) QDP 5.88/2.27 (25) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (26) QDP 5.88/2.27 (27) DependencyGraphProof [EQUIVALENT, 0 ms] 5.88/2.27 (28) QDP 5.88/2.27 (29) UsableRulesProof [EQUIVALENT, 0 ms] 5.88/2.27 (30) QDP 5.88/2.27 (31) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (32) QDP 5.88/2.27 (33) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (34) QDP 5.88/2.27 (35) UsableRulesProof [EQUIVALENT, 0 ms] 5.88/2.27 (36) QDP 5.88/2.27 (37) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (38) QDP 5.88/2.27 (39) DependencyGraphProof [EQUIVALENT, 0 ms] 5.88/2.27 (40) QDP 5.88/2.27 (41) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (42) QDP 5.88/2.27 (43) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (44) QDP 5.88/2.27 (45) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (46) QDP 5.88/2.27 (47) UsableRulesProof [EQUIVALENT, 0 ms] 5.88/2.27 (48) QDP 5.88/2.27 (49) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (50) QDP 5.88/2.27 (51) DependencyGraphProof [EQUIVALENT, 0 ms] 5.88/2.27 (52) QDP 5.88/2.27 (53) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (54) QDP 5.88/2.27 (55) TransformationProof [EQUIVALENT, 0 ms] 5.88/2.27 (56) QDP 5.88/2.27 (57) DependencyGraphProof [EQUIVALENT, 0 ms] 5.88/2.27 (58) QDP 5.88/2.27 (59) MRRProof [EQUIVALENT, 0 ms] 5.88/2.27 (60) QDP 5.88/2.27 (61) PisEmptyProof [EQUIVALENT, 0 ms] 5.88/2.27 (62) YES 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (0) 5.88/2.27 Obligation: 5.88/2.27 Term rewrite system R: 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 f(x, y) -> g(f(y, x)) 5.88/2.27 g(g(g(f(x, y)))) -> x 5.88/2.27 5.88/2.27 5.88/2.27 5.88/2.27 Outermost Strategy. 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (1) Raffelsieper-Zantema-Transformation (SOUND) 5.88/2.27 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (2) 5.88/2.27 Obligation: 5.88/2.27 Q restricted rewrite system: 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 down(g(g(g(f(x, y))))) -> up(x) 5.88/2.27 top(up(x)) -> top(down(x)) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 5.88/2.27 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 Q is empty. 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (3) QTRSRRRProof (EQUIVALENT) 5.88/2.27 Used ordering: 5.88/2.27 Polynomial interpretation [POLO]: 5.88/2.27 5.88/2.27 POL(block(x_1)) = 2 + 2*x_1 5.88/2.27 POL(down(x_1)) = 2*x_1 5.88/2.27 POL(f(x_1, x_2)) = 2*x_1 + 2*x_2 5.88/2.27 POL(f_flat(x_1, x_2)) = 2*x_1 + 2*x_2 5.88/2.27 POL(fresh_constant) = 0 5.88/2.27 POL(g(x_1)) = x_1 5.88/2.27 POL(g_flat(x_1)) = x_1 5.88/2.27 POL(top(x_1)) = 2*x_1 5.88/2.27 POL(up(x_1)) = 2*x_1 5.88/2.27 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 5.88/2.27 5.88/2.27 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 5.88/2.27 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 5.88/2.27 5.88/2.27 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (4) 5.88/2.27 Obligation: 5.88/2.27 Q restricted rewrite system: 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 down(g(g(g(f(x, y))))) -> up(x) 5.88/2.27 top(up(x)) -> top(down(x)) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 Q is empty. 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (5) QTRSRRRProof (EQUIVALENT) 5.88/2.27 Used ordering: 5.88/2.27 Polynomial interpretation [POLO]: 5.88/2.27 5.88/2.27 POL(down(x_1)) = 2*x_1 5.88/2.27 POL(f(x_1, x_2)) = 1 + x_1 + x_2 5.88/2.27 POL(fresh_constant) = 0 5.88/2.27 POL(g(x_1)) = x_1 5.88/2.27 POL(g_flat(x_1)) = x_1 5.88/2.27 POL(top(x_1)) = 2*x_1 5.88/2.27 POL(up(x_1)) = 2*x_1 5.88/2.27 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 5.88/2.27 5.88/2.27 down(g(g(g(f(x, y))))) -> up(x) 5.88/2.27 5.88/2.27 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (6) 5.88/2.27 Obligation: 5.88/2.27 Q restricted rewrite system: 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 top(up(x)) -> top(down(x)) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 Q is empty. 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (7) AAECC Innermost (EQUIVALENT) 5.88/2.27 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 5.88/2.27 The TRS R 2 is 5.88/2.27 top(up(x)) -> top(down(x)) 5.88/2.27 5.88/2.27 The signature Sigma is {top_1} 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (8) 5.88/2.27 Obligation: 5.88/2.27 Q restricted rewrite system: 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 top(up(x)) -> top(down(x)) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 top(up(x0)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (9) DependencyPairsProof (EQUIVALENT) 5.88/2.27 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (10) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(x)) -> TOP(down(x)) 5.88/2.27 TOP(up(x)) -> DOWN(x) 5.88/2.27 DOWN(g(f(y4, y5))) -> G_FLAT(down(f(y4, y5))) 5.88/2.27 DOWN(g(f(y4, y5))) -> DOWN(f(y4, y5)) 5.88/2.27 DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) 5.88/2.27 DOWN(g(fresh_constant)) -> DOWN(fresh_constant) 5.88/2.27 DOWN(g(g(f(y8, y9)))) -> G_FLAT(down(g(f(y8, y9)))) 5.88/2.27 DOWN(g(g(f(y8, y9)))) -> DOWN(g(f(y8, y9))) 5.88/2.27 DOWN(g(g(fresh_constant))) -> G_FLAT(down(g(fresh_constant))) 5.88/2.27 DOWN(g(g(fresh_constant))) -> DOWN(g(fresh_constant)) 5.88/2.27 DOWN(g(g(g(g(y14))))) -> G_FLAT(down(g(g(g(y14))))) 5.88/2.27 DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) 5.88/2.27 DOWN(g(g(g(fresh_constant)))) -> G_FLAT(down(g(g(fresh_constant)))) 5.88/2.27 DOWN(g(g(g(fresh_constant)))) -> DOWN(g(g(fresh_constant))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 top(up(x)) -> top(down(x)) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 top(up(x0)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (11) DependencyGraphProof (EQUIVALENT) 5.88/2.27 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 12 less nodes. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (12) 5.88/2.27 Complex Obligation (AND) 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (13) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 top(up(x)) -> top(down(x)) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 top(up(x0)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (14) UsableRulesProof (EQUIVALENT) 5.88/2.27 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. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (15) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) 5.88/2.27 5.88/2.27 R is empty. 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 top(up(x0)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (16) QReductionProof (EQUIVALENT) 5.88/2.27 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 top(up(x0)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (17) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) 5.88/2.27 5.88/2.27 R is empty. 5.88/2.27 Q is empty. 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (18) QDPSizeChangeProof (EQUIVALENT) 5.88/2.27 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. 5.88/2.27 5.88/2.27 From the DPs we obtained the following set of size-change graphs: 5.88/2.27 *DOWN(g(g(g(g(y14))))) -> DOWN(g(g(g(y14)))) 5.88/2.27 The graph contains the following edges 1 > 1 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (19) 5.88/2.27 YES 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (20) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(x)) -> TOP(down(x)) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 top(up(x)) -> top(down(x)) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 top(up(x0)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (21) UsableRulesProof (EQUIVALENT) 5.88/2.27 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. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (22) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(x)) -> TOP(down(x)) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 top(up(x0)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (23) QReductionProof (EQUIVALENT) 5.88/2.27 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 5.88/2.27 5.88/2.27 top(up(x0)) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (24) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(x)) -> TOP(down(x)) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (25) TransformationProof (EQUIVALENT) 5.88/2.27 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 5.88/2.27 5.88/2.27 (TOP(up(f(x0, x1))) -> TOP(up(g(f(x1, x0)))),TOP(up(f(x0, x1))) -> TOP(up(g(f(x1, x0))))) 5.88/2.27 (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))))) 5.88/2.27 (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) 5.88/2.27 (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)))))) 5.88/2.27 (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))))) 5.88/2.27 (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))))))) 5.88/2.27 (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)))))) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (26) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(f(x0, x1))) -> TOP(up(g(f(x1, x0)))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) 5.88/2.27 TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) 5.88/2.27 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (27) DependencyGraphProof (EQUIVALENT) 5.88/2.27 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (28) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) 5.88/2.27 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(f(y8, y9)))) -> g_flat(down(g(f(y8, y9)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (29) UsableRulesProof (EQUIVALENT) 5.88/2.27 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. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (30) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) 5.88/2.27 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (31) TransformationProof (EQUIVALENT) 5.88/2.27 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]: 5.88/2.27 5.88/2.27 (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)))))) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (32) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(down(g(f(x0, x1))))) 5.88/2.27 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (33) TransformationProof (EQUIVALENT) 5.88/2.27 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]: 5.88/2.27 5.88/2.27 (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)))))) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (34) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 down(g(f(y4, y5))) -> g_flat(down(f(y4, y5))) 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (35) UsableRulesProof (EQUIVALENT) 5.88/2.27 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. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (36) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (37) TransformationProof (EQUIVALENT) 5.88/2.27 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]: 5.88/2.27 5.88/2.27 (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))))) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (38) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) 5.88/2.27 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (39) DependencyGraphProof (EQUIVALENT) 5.88/2.27 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (40) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (41) TransformationProof (EQUIVALENT) 5.88/2.27 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]: 5.88/2.27 5.88/2.27 (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)))))) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (42) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(up(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (43) TransformationProof (EQUIVALENT) 5.88/2.27 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]: 5.88/2.27 5.88/2.27 (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)))))) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (44) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(down(f(x0, x1))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (45) TransformationProof (EQUIVALENT) 5.88/2.27 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]: 5.88/2.27 5.88/2.27 (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))))))) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (46) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 down(f(x, y)) -> up(g(f(y, x))) 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (47) UsableRulesProof (EQUIVALENT) 5.88/2.27 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. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (48) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (49) TransformationProof (EQUIVALENT) 5.88/2.27 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]: 5.88/2.27 5.88/2.27 (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)))))) 5.88/2.27 5.88/2.27 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (50) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) 5.88/2.27 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(g_flat(down(fresh_constant))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (51) DependencyGraphProof (EQUIVALENT) 5.88/2.27 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (52) 5.88/2.27 Obligation: 5.88/2.27 Q DP problem: 5.88/2.27 The TRS P consists of the following rules: 5.88/2.27 5.88/2.27 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.27 TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) 5.88/2.27 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(g_flat(up(g(f(x1, x0)))))) 5.88/2.27 5.88/2.27 The TRS R consists of the following rules: 5.88/2.27 5.88/2.27 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.27 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.27 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.27 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.27 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.27 5.88/2.27 The set Q consists of the following terms: 5.88/2.27 5.88/2.27 down(f(x0, x1)) 5.88/2.27 down(g(f(x0, x1))) 5.88/2.27 down(g(fresh_constant)) 5.88/2.27 down(g(g(f(x0, x1)))) 5.88/2.27 down(g(g(fresh_constant))) 5.88/2.27 down(g(g(g(g(x0))))) 5.88/2.27 down(g(g(g(fresh_constant)))) 5.88/2.27 g_flat(up(x0)) 5.88/2.27 5.88/2.27 We have to consider all minimal (P,Q,R)-chains. 5.88/2.27 ---------------------------------------- 5.88/2.27 5.88/2.27 (53) TransformationProof (EQUIVALENT) 5.88/2.27 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]: 5.88/2.28 5.88/2.28 (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))))))) 5.88/2.28 5.88/2.28 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (54) 5.88/2.28 Obligation: 5.88/2.28 Q DP problem: 5.88/2.28 The TRS P consists of the following rules: 5.88/2.28 5.88/2.28 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.28 TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) 5.88/2.28 TOP(up(g(g(f(x0, x1))))) -> TOP(g_flat(up(g(g(f(x1, x0)))))) 5.88/2.28 5.88/2.28 The TRS R consists of the following rules: 5.88/2.28 5.88/2.28 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.28 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.28 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.28 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.28 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.28 5.88/2.28 The set Q consists of the following terms: 5.88/2.28 5.88/2.28 down(f(x0, x1)) 5.88/2.28 down(g(f(x0, x1))) 5.88/2.28 down(g(fresh_constant)) 5.88/2.28 down(g(g(f(x0, x1)))) 5.88/2.28 down(g(g(fresh_constant))) 5.88/2.28 down(g(g(g(g(x0))))) 5.88/2.28 down(g(g(g(fresh_constant)))) 5.88/2.28 g_flat(up(x0)) 5.88/2.28 5.88/2.28 We have to consider all minimal (P,Q,R)-chains. 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (55) TransformationProof (EQUIVALENT) 5.88/2.28 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]: 5.88/2.28 5.88/2.28 (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))))))) 5.88/2.28 5.88/2.28 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (56) 5.88/2.28 Obligation: 5.88/2.28 Q DP problem: 5.88/2.28 The TRS P consists of the following rules: 5.88/2.28 5.88/2.28 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.28 TOP(up(g(f(x0, x1)))) -> TOP(up(g(g(f(x1, x0))))) 5.88/2.28 TOP(up(g(g(f(x0, x1))))) -> TOP(up(g(g(g(f(x1, x0)))))) 5.88/2.28 5.88/2.28 The TRS R consists of the following rules: 5.88/2.28 5.88/2.28 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.28 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.28 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.28 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.28 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.28 5.88/2.28 The set Q consists of the following terms: 5.88/2.28 5.88/2.28 down(f(x0, x1)) 5.88/2.28 down(g(f(x0, x1))) 5.88/2.28 down(g(fresh_constant)) 5.88/2.28 down(g(g(f(x0, x1)))) 5.88/2.28 down(g(g(fresh_constant))) 5.88/2.28 down(g(g(g(g(x0))))) 5.88/2.28 down(g(g(g(fresh_constant)))) 5.88/2.28 g_flat(up(x0)) 5.88/2.28 5.88/2.28 We have to consider all minimal (P,Q,R)-chains. 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (57) DependencyGraphProof (EQUIVALENT) 5.88/2.28 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (58) 5.88/2.28 Obligation: 5.88/2.28 Q DP problem: 5.88/2.28 The TRS P consists of the following rules: 5.88/2.28 5.88/2.28 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.28 5.88/2.28 The TRS R consists of the following rules: 5.88/2.28 5.88/2.28 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.28 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.28 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.28 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.28 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.28 5.88/2.28 The set Q consists of the following terms: 5.88/2.28 5.88/2.28 down(f(x0, x1)) 5.88/2.28 down(g(f(x0, x1))) 5.88/2.28 down(g(fresh_constant)) 5.88/2.28 down(g(g(f(x0, x1)))) 5.88/2.28 down(g(g(fresh_constant))) 5.88/2.28 down(g(g(g(g(x0))))) 5.88/2.28 down(g(g(g(fresh_constant)))) 5.88/2.28 g_flat(up(x0)) 5.88/2.28 5.88/2.28 We have to consider all minimal (P,Q,R)-chains. 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (59) MRRProof (EQUIVALENT) 5.88/2.28 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. 5.88/2.28 5.88/2.28 Strictly oriented dependency pairs: 5.88/2.28 5.88/2.28 TOP(up(g(g(g(g(x0)))))) -> TOP(g_flat(down(g(g(g(x0)))))) 5.88/2.28 5.88/2.28 Strictly oriented rules of the TRS R: 5.88/2.28 5.88/2.28 g_flat(up(x_1)) -> up(g(x_1)) 5.88/2.28 5.88/2.28 Used ordering: Polynomial interpretation [POLO]: 5.88/2.28 5.88/2.28 POL(TOP(x_1)) = 2*x_1 5.88/2.28 POL(down(x_1)) = x_1 5.88/2.28 POL(fresh_constant) = 0 5.88/2.28 POL(g(x_1)) = 2*x_1 5.88/2.28 POL(g_flat(x_1)) = 2*x_1 5.88/2.28 POL(up(x_1)) = 2 + x_1 5.88/2.28 5.88/2.28 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (60) 5.88/2.28 Obligation: 5.88/2.28 Q DP problem: 5.88/2.28 P is empty. 5.88/2.28 The TRS R consists of the following rules: 5.88/2.28 5.88/2.28 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 5.88/2.28 down(g(g(g(g(y14))))) -> g_flat(down(g(g(g(y14))))) 5.88/2.28 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 5.88/2.28 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 5.88/2.28 5.88/2.28 The set Q consists of the following terms: 5.88/2.28 5.88/2.28 down(f(x0, x1)) 5.88/2.28 down(g(f(x0, x1))) 5.88/2.28 down(g(fresh_constant)) 5.88/2.28 down(g(g(f(x0, x1)))) 5.88/2.28 down(g(g(fresh_constant))) 5.88/2.28 down(g(g(g(g(x0))))) 5.88/2.28 down(g(g(g(fresh_constant)))) 5.88/2.28 g_flat(up(x0)) 5.88/2.28 5.88/2.28 We have to consider all minimal (P,Q,R)-chains. 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (61) PisEmptyProof (EQUIVALENT) 5.88/2.28 The TRS P is empty. Hence, there is no (P,Q,R) chain. 5.88/2.28 ---------------------------------------- 5.88/2.28 5.88/2.28 (62) 5.88/2.28 YES 6.10/2.30 EOF