178.67/77.06 MAYBE 178.67/77.07 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 178.67/77.07 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 178.67/77.07 178.67/77.07 178.67/77.07 Outermost Termination of the given OTRS could not be shown: 178.67/77.07 178.67/77.07 (0) OTRS 178.67/77.07 (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] 178.67/77.07 (2) QTRS 178.67/77.07 (3) DependencyPairsProof [EQUIVALENT, 2 ms] 178.67/77.07 (4) QDP 178.67/77.07 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 178.67/77.07 (6) AND 178.67/77.07 (7) QDP 178.67/77.07 (8) UsableRulesProof [EQUIVALENT, 0 ms] 178.67/77.07 (9) QDP 178.67/77.07 (10) QReductionProof [EQUIVALENT, 0 ms] 178.67/77.07 (11) QDP 178.67/77.07 (12) MRRProof [EQUIVALENT, 0 ms] 178.67/77.07 (13) QDP 178.67/77.07 (14) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 178.67/77.07 (15) QDP 178.67/77.07 (16) UsableRulesReductionPairsProof [EQUIVALENT, 9 ms] 178.67/77.07 (17) QDP 178.67/77.07 (18) DependencyGraphProof [EQUIVALENT, 0 ms] 178.67/77.07 (19) TRUE 178.67/77.07 (20) QDP 178.67/77.07 (21) UsableRulesProof [EQUIVALENT, 0 ms] 178.67/77.07 (22) QDP 178.67/77.07 (23) QReductionProof [EQUIVALENT, 0 ms] 178.67/77.07 (24) QDP 178.67/77.07 (25) TransformationProof [EQUIVALENT, 0 ms] 178.67/77.07 (26) QDP 178.67/77.07 (27) UsableRulesProof [EQUIVALENT, 0 ms] 178.67/77.07 (28) QDP 178.67/77.07 (29) QReductionProof [EQUIVALENT, 0 ms] 178.67/77.07 (30) QDP 178.67/77.07 (31) Trivial-Transformation [SOUND, 0 ms] 178.67/77.07 (32) QTRS 178.67/77.07 (33) DependencyPairsProof [EQUIVALENT, 0 ms] 178.67/77.07 (34) QDP 178.67/77.07 (35) DependencyGraphProof [EQUIVALENT, 0 ms] 178.67/77.07 (36) QDP 178.67/77.07 (37) NonTerminationLoopProof [COMPLETE, 0 ms] 178.67/77.07 (38) NO 178.67/77.07 (39) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 178.67/77.07 (40) QTRS 178.67/77.07 (41) DependencyPairsProof [EQUIVALENT, 0 ms] 178.67/77.07 (42) QDP 178.67/77.07 (43) DependencyGraphProof [EQUIVALENT, 0 ms] 178.67/77.07 (44) AND 178.67/77.07 (45) QDP 178.67/77.07 (46) UsableRulesProof [EQUIVALENT, 0 ms] 178.67/77.07 (47) QDP 178.67/77.07 (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] 178.67/77.07 (49) YES 178.67/77.07 (50) QDP 178.67/77.07 (51) TransformationProof [EQUIVALENT, 0 ms] 178.67/77.07 (52) QDP 178.67/77.07 (53) DependencyGraphProof [EQUIVALENT, 0 ms] 178.67/77.07 (54) QDP 178.67/77.07 (55) QDPOrderProof [EQUIVALENT, 17 ms] 178.67/77.07 (56) QDP 178.67/77.07 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (0) 178.67/77.07 Obligation: 178.67/77.07 Term rewrite system R: 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 f(x, c) -> g(f(x, x)) 178.67/77.07 g(g(x)) -> c 178.67/77.07 178.67/77.07 178.67/77.07 178.67/77.07 Outermost Strategy. 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (1) Thiemann-SpecialC-Transformation (EQUIVALENT) 178.67/77.07 We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (2) 178.67/77.07 Obligation: 178.67/77.07 Q restricted rewrite system: 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 top(go_up(x)) -> top(reduce(x)) 178.67/77.07 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.07 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 check_f(result_f(x)) -> go_up(x) 178.67/77.07 check_g(result_g(x)) -> go_up(x) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.07 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.07 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.07 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.07 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 top(go_up(x0)) 178.67/77.07 reduce(f(x0, x1)) 178.67/77.07 reduce(g(x0)) 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 check_f(result_f(x0)) 178.67/77.07 check_g(result_g(x0)) 178.67/77.07 check_f(redex_f(x0, x1)) 178.67/77.07 check_g(redex_g(x0)) 178.67/77.07 in_f_1(go_up(x0), x1) 178.67/77.07 in_f_2(x0, go_up(x1)) 178.67/77.07 in_g_1(go_up(x0)) 178.67/77.07 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (3) DependencyPairsProof (EQUIVALENT) 178.67/77.07 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (4) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 TOP(go_up(x)) -> TOP(reduce(x)) 178.67/77.07 TOP(go_up(x)) -> REDUCE(x) 178.67/77.07 REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) 178.67/77.07 REDUCE(f(x_1, x_2)) -> REDEX_F(x_1, x_2) 178.67/77.07 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.67/77.07 REDUCE(g(x_1)) -> REDEX_G(x_1) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> IN_F_1(reduce(x_1), x_2) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> IN_F_2(x_1, reduce(x_2)) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) 178.67/77.07 CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1)) 178.67/77.07 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.67/77.07 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 top(go_up(x)) -> top(reduce(x)) 178.67/77.07 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.07 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 check_f(result_f(x)) -> go_up(x) 178.67/77.07 check_g(result_g(x)) -> go_up(x) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.07 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.07 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.07 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.07 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 top(go_up(x0)) 178.67/77.07 reduce(f(x0, x1)) 178.67/77.07 reduce(g(x0)) 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 check_f(result_f(x0)) 178.67/77.07 check_g(result_g(x0)) 178.67/77.07 check_f(redex_f(x0, x1)) 178.67/77.07 check_g(redex_g(x0)) 178.67/77.07 in_f_1(go_up(x0), x1) 178.67/77.07 in_f_2(x0, go_up(x1)) 178.67/77.07 in_g_1(go_up(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (5) DependencyGraphProof (EQUIVALENT) 178.67/77.07 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (6) 178.67/77.07 Complex Obligation (AND) 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (7) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) 178.67/77.07 REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) 178.67/77.07 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.67/77.07 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.67/77.07 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 top(go_up(x)) -> top(reduce(x)) 178.67/77.07 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.07 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 check_f(result_f(x)) -> go_up(x) 178.67/77.07 check_g(result_g(x)) -> go_up(x) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.07 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.07 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.07 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.07 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 top(go_up(x0)) 178.67/77.07 reduce(f(x0, x1)) 178.67/77.07 reduce(g(x0)) 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 check_f(result_f(x0)) 178.67/77.07 check_g(result_g(x0)) 178.67/77.07 check_f(redex_f(x0, x1)) 178.67/77.07 check_g(redex_g(x0)) 178.67/77.07 in_f_1(go_up(x0), x1) 178.67/77.07 in_f_2(x0, go_up(x1)) 178.67/77.07 in_g_1(go_up(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (8) UsableRulesProof (EQUIVALENT) 178.67/77.07 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. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (9) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) 178.67/77.07 REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) 178.67/77.07 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.67/77.07 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.67/77.07 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 top(go_up(x0)) 178.67/77.07 reduce(f(x0, x1)) 178.67/77.07 reduce(g(x0)) 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 check_f(result_f(x0)) 178.67/77.07 check_g(result_g(x0)) 178.67/77.07 check_f(redex_f(x0, x1)) 178.67/77.07 check_g(redex_g(x0)) 178.67/77.07 in_f_1(go_up(x0), x1) 178.67/77.07 in_f_2(x0, go_up(x1)) 178.67/77.07 in_g_1(go_up(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (10) QReductionProof (EQUIVALENT) 178.67/77.07 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.67/77.07 178.67/77.07 top(go_up(x0)) 178.67/77.07 reduce(f(x0, x1)) 178.67/77.07 reduce(g(x0)) 178.67/77.07 check_f(result_f(x0)) 178.67/77.07 check_g(result_g(x0)) 178.67/77.07 check_f(redex_f(x0, x1)) 178.67/77.07 check_g(redex_g(x0)) 178.67/77.07 in_f_1(go_up(x0), x1) 178.67/77.07 in_f_2(x0, go_up(x1)) 178.67/77.07 in_g_1(go_up(x0)) 178.67/77.07 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (11) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) 178.67/77.07 REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) 178.67/77.07 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.67/77.07 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.67/77.07 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (12) MRRProof (EQUIVALENT) 178.67/77.07 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. 178.67/77.07 178.67/77.07 178.67/77.07 Strictly oriented rules of the TRS R: 178.67/77.07 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 178.67/77.07 Used ordering: Polynomial interpretation [POLO]: 178.67/77.07 178.67/77.07 POL(CHECK_F(x_1)) = 2 + x_1 178.67/77.07 POL(CHECK_G(x_1)) = x_1 178.67/77.07 POL(REDUCE(x_1)) = 2 + 2*x_1 178.67/77.07 POL(c) = 0 178.67/77.07 POL(f(x_1, x_2)) = x_1 + x_2 178.67/77.07 POL(g(x_1)) = x_1 178.67/77.07 POL(redex_f(x_1, x_2)) = 2*x_1 + 2*x_2 178.67/77.07 POL(redex_g(x_1)) = 2 + 2*x_1 178.67/77.07 POL(result_f(x_1)) = x_1 178.67/77.07 POL(result_g(x_1)) = 1 + 2*x_1 178.67/77.07 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (13) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) 178.67/77.07 REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) 178.67/77.07 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.67/77.07 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.67/77.07 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (14) UsableRulesReductionPairsProof (EQUIVALENT) 178.67/77.07 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. 178.67/77.07 178.67/77.07 No dependency pairs are removed. 178.67/77.07 178.67/77.07 The following rules are removed from R: 178.67/77.07 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 Used ordering: POLO with Polynomial interpretation [POLO]: 178.67/77.07 178.67/77.07 POL(CHECK_F(x_1)) = x_1 178.67/77.07 POL(CHECK_G(x_1)) = 2*x_1 178.67/77.07 POL(REDUCE(x_1)) = 2*x_1 178.67/77.07 POL(c) = 0 178.67/77.07 POL(f(x_1, x_2)) = x_1 + x_2 178.67/77.07 POL(g(x_1)) = x_1 178.67/77.07 POL(redex_f(x_1, x_2)) = 2*x_1 + 2*x_2 178.67/77.07 POL(redex_g(x_1)) = x_1 178.67/77.07 POL(result_f(x_1)) = x_1 178.67/77.07 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (15) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) 178.67/77.07 REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) 178.67/77.07 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.67/77.07 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.67/77.07 178.67/77.07 R is empty. 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (16) UsableRulesReductionPairsProof (EQUIVALENT) 178.67/77.07 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. 178.67/77.07 178.67/77.07 The following dependency pairs can be deleted: 178.67/77.07 178.67/77.07 REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2)) 178.67/77.07 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.67/77.07 No rules are removed from R. 178.67/77.07 178.67/77.07 Used ordering: POLO with Polynomial interpretation [POLO]: 178.67/77.07 178.67/77.07 POL(CHECK_F(x_1)) = x_1 178.67/77.07 POL(CHECK_G(x_1)) = 2*x_1 178.67/77.07 POL(REDUCE(x_1)) = 2*x_1 178.67/77.07 POL(f(x_1, x_2)) = 2*x_1 + 2*x_2 178.67/77.07 POL(g(x_1)) = 2*x_1 178.67/77.07 POL(redex_f(x_1, x_2)) = 2*x_1 + 2*x_2 178.67/77.07 POL(redex_g(x_1)) = x_1 178.67/77.07 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (17) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1) 178.67/77.07 CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2) 178.67/77.07 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.67/77.07 178.67/77.07 R is empty. 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (18) DependencyGraphProof (EQUIVALENT) 178.67/77.07 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (19) 178.67/77.07 TRUE 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (20) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 TOP(go_up(x)) -> TOP(reduce(x)) 178.67/77.07 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 top(go_up(x)) -> top(reduce(x)) 178.67/77.07 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.07 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 check_f(result_f(x)) -> go_up(x) 178.67/77.07 check_g(result_g(x)) -> go_up(x) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.07 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.07 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.07 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.07 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 top(go_up(x0)) 178.67/77.07 reduce(f(x0, x1)) 178.67/77.07 reduce(g(x0)) 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 check_f(result_f(x0)) 178.67/77.07 check_g(result_g(x0)) 178.67/77.07 check_f(redex_f(x0, x1)) 178.67/77.07 check_g(redex_g(x0)) 178.67/77.07 in_f_1(go_up(x0), x1) 178.67/77.07 in_f_2(x0, go_up(x1)) 178.67/77.07 in_g_1(go_up(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (21) UsableRulesProof (EQUIVALENT) 178.67/77.07 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. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (22) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 TOP(go_up(x)) -> TOP(reduce(x)) 178.67/77.07 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.07 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 check_g(result_g(x)) -> go_up(x) 178.67/77.07 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.07 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 check_f(result_f(x)) -> go_up(x) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.07 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.07 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 top(go_up(x0)) 178.67/77.07 reduce(f(x0, x1)) 178.67/77.07 reduce(g(x0)) 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 check_f(result_f(x0)) 178.67/77.07 check_g(result_g(x0)) 178.67/77.07 check_f(redex_f(x0, x1)) 178.67/77.07 check_g(redex_g(x0)) 178.67/77.07 in_f_1(go_up(x0), x1) 178.67/77.07 in_f_2(x0, go_up(x1)) 178.67/77.07 in_g_1(go_up(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (23) QReductionProof (EQUIVALENT) 178.67/77.07 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.67/77.07 178.67/77.07 top(go_up(x0)) 178.67/77.07 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (24) 178.67/77.07 Obligation: 178.67/77.07 Q DP problem: 178.67/77.07 The TRS P consists of the following rules: 178.67/77.07 178.67/77.07 TOP(go_up(x)) -> TOP(reduce(x)) 178.67/77.07 178.67/77.07 The TRS R consists of the following rules: 178.67/77.07 178.67/77.07 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.07 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.07 redex_g(g(x)) -> result_g(c) 178.67/77.07 check_g(result_g(x)) -> go_up(x) 178.67/77.07 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.07 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.07 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.07 check_f(result_f(x)) -> go_up(x) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.07 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.07 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.07 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.07 178.67/77.07 The set Q consists of the following terms: 178.67/77.07 178.67/77.07 reduce(f(x0, x1)) 178.67/77.07 reduce(g(x0)) 178.67/77.07 redex_f(x0, c) 178.67/77.07 redex_g(g(x0)) 178.67/77.07 check_f(result_f(x0)) 178.67/77.07 check_g(result_g(x0)) 178.67/77.07 check_f(redex_f(x0, x1)) 178.67/77.07 check_g(redex_g(x0)) 178.67/77.07 in_f_1(go_up(x0), x1) 178.67/77.07 in_f_2(x0, go_up(x1)) 178.67/77.07 in_g_1(go_up(x0)) 178.67/77.07 178.67/77.07 We have to consider all minimal (P,Q,R)-chains. 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (25) TransformationProof (EQUIVALENT) 178.67/77.07 By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: 178.67/77.07 178.67/77.07 (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)))) 178.67/77.07 (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))) 178.67/77.07 178.67/77.07 178.67/77.07 ---------------------------------------- 178.67/77.07 178.67/77.07 (26) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 TOP(go_up(f(x0, x1))) -> TOP(check_f(redex_f(x0, x1))) 178.67/77.08 TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.08 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.08 redex_g(g(x)) -> result_g(c) 178.67/77.08 check_g(result_g(x)) -> go_up(x) 178.67/77.08 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.08 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.08 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.08 check_f(result_f(x)) -> go_up(x) 178.67/77.08 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.08 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.08 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.08 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.08 178.67/77.08 The set Q consists of the following terms: 178.67/77.08 178.67/77.08 reduce(f(x0, x1)) 178.67/77.08 reduce(g(x0)) 178.67/77.08 redex_f(x0, c) 178.67/77.08 redex_g(g(x0)) 178.67/77.08 check_f(result_f(x0)) 178.67/77.08 check_g(result_g(x0)) 178.67/77.08 check_f(redex_f(x0, x1)) 178.67/77.08 check_g(redex_g(x0)) 178.67/77.08 in_f_1(go_up(x0), x1) 178.67/77.08 in_f_2(x0, go_up(x1)) 178.67/77.08 in_g_1(go_up(x0)) 178.67/77.08 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (27) UsableRulesProof (EQUIVALENT) 178.67/77.08 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. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (28) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 TOP(go_up(x)) -> TOP(reduce(x)) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.08 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.08 redex_g(g(x)) -> result_g(c) 178.67/77.08 check_g(result_g(x)) -> go_up(x) 178.67/77.08 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.08 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.08 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.08 check_f(result_f(x)) -> go_up(x) 178.67/77.08 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.08 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.08 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.08 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.08 178.67/77.08 The set Q consists of the following terms: 178.67/77.08 178.67/77.08 top(go_up(x0)) 178.67/77.08 reduce(f(x0, x1)) 178.67/77.08 reduce(g(x0)) 178.67/77.08 redex_f(x0, c) 178.67/77.08 redex_g(g(x0)) 178.67/77.08 check_f(result_f(x0)) 178.67/77.08 check_g(result_g(x0)) 178.67/77.08 check_f(redex_f(x0, x1)) 178.67/77.08 check_g(redex_g(x0)) 178.67/77.08 in_f_1(go_up(x0), x1) 178.67/77.08 in_f_2(x0, go_up(x1)) 178.67/77.08 in_g_1(go_up(x0)) 178.67/77.08 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (29) QReductionProof (EQUIVALENT) 178.67/77.08 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.67/77.08 178.67/77.08 top(go_up(x0)) 178.67/77.08 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (30) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 TOP(go_up(x)) -> TOP(reduce(x)) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2)) 178.67/77.08 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.67/77.08 redex_g(g(x)) -> result_g(c) 178.67/77.08 check_g(result_g(x)) -> go_up(x) 178.67/77.08 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.67/77.08 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.67/77.08 redex_f(x, c) -> result_f(g(f(x, x))) 178.67/77.08 check_f(result_f(x)) -> go_up(x) 178.67/77.08 check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2) 178.67/77.08 check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2)) 178.67/77.08 in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2)) 178.67/77.08 in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2)) 178.67/77.08 178.67/77.08 The set Q consists of the following terms: 178.67/77.08 178.67/77.08 reduce(f(x0, x1)) 178.67/77.08 reduce(g(x0)) 178.67/77.08 redex_f(x0, c) 178.67/77.08 redex_g(g(x0)) 178.67/77.08 check_f(result_f(x0)) 178.67/77.08 check_g(result_g(x0)) 178.67/77.08 check_f(redex_f(x0, x1)) 178.67/77.08 check_g(redex_g(x0)) 178.67/77.08 in_f_1(go_up(x0), x1) 178.67/77.08 in_f_2(x0, go_up(x1)) 178.67/77.08 in_g_1(go_up(x0)) 178.67/77.08 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (31) Trivial-Transformation (SOUND) 178.67/77.08 We applied the Trivial transformation to transform the outermost TRS to a standard TRS. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (32) 178.67/77.08 Obligation: 178.67/77.08 Q restricted rewrite system: 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 f(x, c) -> g(f(x, x)) 178.67/77.08 g(g(x)) -> c 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (33) DependencyPairsProof (EQUIVALENT) 178.67/77.08 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (34) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 F(x, c) -> G(f(x, x)) 178.67/77.08 F(x, c) -> F(x, x) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 f(x, c) -> g(f(x, x)) 178.67/77.08 g(g(x)) -> c 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (35) DependencyGraphProof (EQUIVALENT) 178.67/77.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (36) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 F(x, c) -> F(x, x) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 f(x, c) -> g(f(x, x)) 178.67/77.08 g(g(x)) -> c 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (37) NonTerminationLoopProof (COMPLETE) 178.67/77.08 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 178.67/77.08 Found a loop by semiunifying a rule from P directly. 178.67/77.08 178.67/77.08 s = F(x, c) evaluates to t =F(x, x) 178.67/77.08 178.67/77.08 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 178.67/77.08 * Matcher: [ ] 178.67/77.08 * Semiunifier: [x / c] 178.67/77.08 178.67/77.08 -------------------------------------------------------------------------------- 178.67/77.08 Rewriting sequence 178.67/77.08 178.67/77.08 The DP semiunifies directly so there is only one rewrite step from F(c, c) to F(c, c). 178.67/77.08 178.67/77.08 178.67/77.08 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (38) 178.67/77.08 NO 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (39) Raffelsieper-Zantema-Transformation (SOUND) 178.67/77.08 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (40) 178.67/77.08 Obligation: 178.67/77.08 Q restricted rewrite system: 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 down(f(x, c)) -> up(g(f(x, x))) 178.67/77.08 down(g(g(x))) -> up(c) 178.67/77.08 top(up(x)) -> top(down(x)) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) 178.67/77.08 down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) 178.67/77.08 down(g(c)) -> g_flat(down(c)) 178.67/77.08 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.67/77.08 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 g_flat(up(x_1)) -> up(g(x_1)) 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (41) DependencyPairsProof (EQUIVALENT) 178.67/77.08 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (42) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 TOP(up(x)) -> TOP(down(x)) 178.67/77.08 TOP(up(x)) -> DOWN(x) 178.67/77.08 DOWN(f(y12, f(y13, y14))) -> F_FLAT(down(y12), block(f(y13, y14))) 178.67/77.08 DOWN(f(y12, f(y13, y14))) -> DOWN(y12) 178.67/77.08 DOWN(f(y12, f(y13, y14))) -> F_FLAT(block(y12), down(f(y13, y14))) 178.67/77.08 DOWN(f(y12, f(y13, y14))) -> DOWN(f(y13, y14)) 178.67/77.08 DOWN(f(y30, g(y31))) -> F_FLAT(down(y30), block(g(y31))) 178.67/77.08 DOWN(f(y30, g(y31))) -> DOWN(y30) 178.67/77.08 DOWN(f(y30, g(y31))) -> F_FLAT(block(y30), down(g(y31))) 178.67/77.08 DOWN(f(y30, g(y31))) -> DOWN(g(y31)) 178.67/77.08 DOWN(f(y39, fresh_constant)) -> F_FLAT(down(y39), block(fresh_constant)) 178.67/77.08 DOWN(f(y39, fresh_constant)) -> DOWN(y39) 178.67/77.08 DOWN(f(y39, fresh_constant)) -> F_FLAT(block(y39), down(fresh_constant)) 178.67/77.08 DOWN(f(y39, fresh_constant)) -> DOWN(fresh_constant) 178.67/77.08 DOWN(g(f(y44, y45))) -> G_FLAT(down(f(y44, y45))) 178.67/77.08 DOWN(g(f(y44, y45))) -> DOWN(f(y44, y45)) 178.67/77.08 DOWN(g(c)) -> G_FLAT(down(c)) 178.67/77.08 DOWN(g(c)) -> DOWN(c) 178.67/77.08 DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) 178.67/77.08 DOWN(g(fresh_constant)) -> DOWN(fresh_constant) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 down(f(x, c)) -> up(g(f(x, x))) 178.67/77.08 down(g(g(x))) -> up(c) 178.67/77.08 top(up(x)) -> top(down(x)) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) 178.67/77.08 down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) 178.67/77.08 down(g(c)) -> g_flat(down(c)) 178.67/77.08 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.67/77.08 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 g_flat(up(x_1)) -> up(g(x_1)) 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (43) DependencyGraphProof (EQUIVALENT) 178.67/77.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 13 less nodes. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (44) 178.67/77.08 Complex Obligation (AND) 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (45) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 DOWN(f(y12, f(y13, y14))) -> DOWN(f(y13, y14)) 178.67/77.08 DOWN(f(y12, f(y13, y14))) -> DOWN(y12) 178.67/77.08 DOWN(f(y30, g(y31))) -> DOWN(y30) 178.67/77.08 DOWN(f(y30, g(y31))) -> DOWN(g(y31)) 178.67/77.08 DOWN(g(f(y44, y45))) -> DOWN(f(y44, y45)) 178.67/77.08 DOWN(f(y39, fresh_constant)) -> DOWN(y39) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 down(f(x, c)) -> up(g(f(x, x))) 178.67/77.08 down(g(g(x))) -> up(c) 178.67/77.08 top(up(x)) -> top(down(x)) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) 178.67/77.08 down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) 178.67/77.08 down(g(c)) -> g_flat(down(c)) 178.67/77.08 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.67/77.08 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 g_flat(up(x_1)) -> up(g(x_1)) 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (46) UsableRulesProof (EQUIVALENT) 178.67/77.08 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. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (47) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 DOWN(f(y12, f(y13, y14))) -> DOWN(f(y13, y14)) 178.67/77.08 DOWN(f(y12, f(y13, y14))) -> DOWN(y12) 178.67/77.08 DOWN(f(y30, g(y31))) -> DOWN(y30) 178.67/77.08 DOWN(f(y30, g(y31))) -> DOWN(g(y31)) 178.67/77.08 DOWN(g(f(y44, y45))) -> DOWN(f(y44, y45)) 178.67/77.08 DOWN(f(y39, fresh_constant)) -> DOWN(y39) 178.67/77.08 178.67/77.08 R is empty. 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (48) QDPSizeChangeProof (EQUIVALENT) 178.67/77.08 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. 178.67/77.08 178.67/77.08 From the DPs we obtained the following set of size-change graphs: 178.67/77.08 *DOWN(f(y30, g(y31))) -> DOWN(g(y31)) 178.67/77.08 The graph contains the following edges 1 > 1 178.67/77.08 178.67/77.08 178.67/77.08 *DOWN(g(f(y44, y45))) -> DOWN(f(y44, y45)) 178.67/77.08 The graph contains the following edges 1 > 1 178.67/77.08 178.67/77.08 178.67/77.08 *DOWN(f(y12, f(y13, y14))) -> DOWN(f(y13, y14)) 178.67/77.08 The graph contains the following edges 1 > 1 178.67/77.08 178.67/77.08 178.67/77.08 *DOWN(f(y12, f(y13, y14))) -> DOWN(y12) 178.67/77.08 The graph contains the following edges 1 > 1 178.67/77.08 178.67/77.08 178.67/77.08 *DOWN(f(y30, g(y31))) -> DOWN(y30) 178.67/77.08 The graph contains the following edges 1 > 1 178.67/77.08 178.67/77.08 178.67/77.08 *DOWN(f(y39, fresh_constant)) -> DOWN(y39) 178.67/77.08 The graph contains the following edges 1 > 1 178.67/77.08 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (49) 178.67/77.08 YES 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (50) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 TOP(up(x)) -> TOP(down(x)) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 down(f(x, c)) -> up(g(f(x, x))) 178.67/77.08 down(g(g(x))) -> up(c) 178.67/77.08 top(up(x)) -> top(down(x)) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) 178.67/77.08 down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) 178.67/77.08 down(g(c)) -> g_flat(down(c)) 178.67/77.08 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.67/77.08 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 g_flat(up(x_1)) -> up(g(x_1)) 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (51) TransformationProof (EQUIVALENT) 178.67/77.08 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 178.67/77.08 178.67/77.08 (TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0)))),TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0))))) 178.67/77.08 (TOP(up(g(g(x0)))) -> TOP(up(c)),TOP(up(g(g(x0)))) -> TOP(up(c))) 178.67/77.08 (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))))) 178.67/77.08 (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))))) 178.67/77.08 (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))))) 178.67/77.08 (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))))) 178.67/77.08 (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)))) 178.67/77.08 (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)))) 178.67/77.08 (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))))) 178.67/77.08 (TOP(up(g(c))) -> TOP(g_flat(down(c))),TOP(up(g(c))) -> TOP(g_flat(down(c)))) 178.67/77.08 (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) 178.67/77.08 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (52) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0)))) 178.67/77.08 TOP(up(g(g(x0)))) -> TOP(up(c)) 178.67/77.08 TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(down(x0), block(f(x1, x2)))) 178.67/77.08 TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(block(x0), down(f(x1, x2)))) 178.67/77.08 TOP(up(f(x0, g(x1)))) -> TOP(f_flat(down(x0), block(g(x1)))) 178.67/77.08 TOP(up(f(x0, g(x1)))) -> TOP(f_flat(block(x0), down(g(x1)))) 178.67/77.08 TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(down(x0), block(fresh_constant))) 178.67/77.08 TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(block(x0), down(fresh_constant))) 178.67/77.08 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) 178.67/77.08 TOP(up(g(c))) -> TOP(g_flat(down(c))) 178.67/77.08 TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 down(f(x, c)) -> up(g(f(x, x))) 178.67/77.08 down(g(g(x))) -> up(c) 178.67/77.08 top(up(x)) -> top(down(x)) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) 178.67/77.08 down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) 178.67/77.08 down(g(c)) -> g_flat(down(c)) 178.67/77.08 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.67/77.08 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 g_flat(up(x_1)) -> up(g(x_1)) 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (53) DependencyGraphProof (EQUIVALENT) 178.67/77.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (54) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) 178.67/77.08 TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0)))) 178.67/77.08 TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(down(x0), block(f(x1, x2)))) 178.67/77.08 TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(block(x0), down(f(x1, x2)))) 178.67/77.08 TOP(up(f(x0, g(x1)))) -> TOP(f_flat(down(x0), block(g(x1)))) 178.67/77.08 TOP(up(f(x0, g(x1)))) -> TOP(f_flat(block(x0), down(g(x1)))) 178.67/77.08 TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(down(x0), block(fresh_constant))) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 down(f(x, c)) -> up(g(f(x, x))) 178.67/77.08 down(g(g(x))) -> up(c) 178.67/77.08 top(up(x)) -> top(down(x)) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) 178.67/77.08 down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) 178.67/77.08 down(g(c)) -> g_flat(down(c)) 178.67/77.08 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.67/77.08 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 g_flat(up(x_1)) -> up(g(x_1)) 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (55) QDPOrderProof (EQUIVALENT) 178.67/77.08 We use the reduction pair processor [LPAR04,JAR06]. 178.67/77.08 178.67/77.08 178.67/77.08 The following pairs can be oriented strictly and are deleted. 178.67/77.08 178.67/77.08 TOP(up(f(x0, c))) -> TOP(up(g(f(x0, x0)))) 178.67/77.08 The remaining pairs can at least be oriented weakly. 178.67/77.08 Used ordering: Polynomial interpretation [POLO]: 178.67/77.08 178.67/77.08 POL(TOP(x_1)) = x_1 178.67/77.08 POL(block(x_1)) = 0 178.67/77.08 POL(c) = 0 178.67/77.08 POL(down(x_1)) = 0 178.67/77.08 POL(f(x_1, x_2)) = 1 178.67/77.08 POL(f_flat(x_1, x_2)) = 1 178.67/77.08 POL(fresh_constant) = 0 178.67/77.08 POL(g(x_1)) = 0 178.67/77.08 POL(g_flat(x_1)) = 0 178.67/77.08 POL(up(x_1)) = x_1 178.67/77.08 178.67/77.08 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 178.67/77.08 178.67/77.08 g_flat(up(x_1)) -> up(g(x_1)) 178.67/77.08 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 178.67/77.08 178.67/77.08 ---------------------------------------- 178.67/77.08 178.67/77.08 (56) 178.67/77.08 Obligation: 178.67/77.08 Q DP problem: 178.67/77.08 The TRS P consists of the following rules: 178.67/77.08 178.67/77.08 TOP(up(g(f(x0, x1)))) -> TOP(g_flat(down(f(x0, x1)))) 178.67/77.08 TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(down(x0), block(f(x1, x2)))) 178.67/77.08 TOP(up(f(x0, f(x1, x2)))) -> TOP(f_flat(block(x0), down(f(x1, x2)))) 178.67/77.08 TOP(up(f(x0, g(x1)))) -> TOP(f_flat(down(x0), block(g(x1)))) 178.67/77.08 TOP(up(f(x0, g(x1)))) -> TOP(f_flat(block(x0), down(g(x1)))) 178.67/77.08 TOP(up(f(x0, fresh_constant))) -> TOP(f_flat(down(x0), block(fresh_constant))) 178.67/77.08 178.67/77.08 The TRS R consists of the following rules: 178.67/77.08 178.67/77.08 down(f(x, c)) -> up(g(f(x, x))) 178.67/77.08 down(g(g(x))) -> up(c) 178.67/77.08 top(up(x)) -> top(down(x)) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(down(y12), block(f(y13, y14))) 178.67/77.08 down(f(y12, f(y13, y14))) -> f_flat(block(y12), down(f(y13, y14))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(down(y30), block(g(y31))) 178.67/77.08 down(f(y30, g(y31))) -> f_flat(block(y30), down(g(y31))) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(down(y39), block(fresh_constant)) 178.67/77.08 down(f(y39, fresh_constant)) -> f_flat(block(y39), down(fresh_constant)) 178.67/77.08 down(g(f(y44, y45))) -> g_flat(down(f(y44, y45))) 178.67/77.08 down(g(c)) -> g_flat(down(c)) 178.67/77.08 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.67/77.08 f_flat(up(x_1), block(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 f_flat(block(x_1), up(x_2)) -> up(f(x_1, x_2)) 178.67/77.08 g_flat(up(x_1)) -> up(g(x_1)) 178.67/77.08 178.67/77.08 Q is empty. 178.67/77.08 We have to consider all minimal (P,Q,R)-chains. 178.82/77.13 EOF