/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) Overlay + Local Confluence [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 216 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) QDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) QDP (10) ATransformationProof [EQUIVALENT, 0 ms] (11) QDP (12) QReductionProof [EQUIVALENT, 0 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) QDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) QDP (19) ATransformationProof [EQUIVALENT, 0 ms] (20) QDP (21) QReductionProof [EQUIVALENT, 0 ms] (22) QDP (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] (24) YES (25) QDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) QDP (28) ATransformationProof [EQUIVALENT, 0 ms] (29) QDP (30) QReductionProof [EQUIVALENT, 0 ms] (31) QDP (32) QDPOrderProof [EQUIVALENT, 52 ms] (33) QDP (34) DependencyGraphProof [EQUIVALENT, 0 ms] (35) QDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) QDP (38) QReductionProof [EQUIVALENT, 0 ms] (39) QDP (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] (41) YES (42) QDP (43) QDPSizeChangeProof [EQUIVALENT, 0 ms] (44) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: app(app(eq, 0), 0) -> true app(app(eq, 0), app(s, x)) -> false app(app(eq, app(s, x)), 0) -> false app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y) app(app(or, true), y) -> true app(app(or, false), y) -> y app(app(union, empty), h) -> h app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h)) app(app(app(app(reach, x), y), empty), h) -> false app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) -> true app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) Q is empty. ---------------------------------------- (1) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: app(app(eq, 0), 0) -> true app(app(eq, 0), app(s, x)) -> false app(app(eq, app(s, x)), 0) -> false app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y) app(app(or, true), y) -> true app(app(or, false), y) -> y app(app(union, empty), h) -> h app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h)) app(app(app(app(reach, x), y), empty), h) -> false app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) -> true app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y) APP(app(eq, app(s, x)), app(s, y)) -> APP(eq, x) APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(app(app(edge, x), y), app(app(union, i), h)) APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(app(union, i), h) APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(union, i) APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)) APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(if_reach_1, app(app(eq, x), u)), x), y) APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(if_reach_1, app(app(eq, x), u)), x) APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(if_reach_1, app(app(eq, x), u)) APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(eq, x), u) APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(eq, x) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(if_reach_2, app(app(eq, y), v)), x), y) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(if_reach_2, app(app(eq, y), v)), x) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(if_reach_2, app(app(eq, y), v)) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(eq, y), v) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(eq, y) APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, x), y), i) APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, x), y) APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(reach, x) APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(edge, u), v), h) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty)) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(or, app(app(app(app(reach, x), y), i), h)) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), h) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, x), y), i) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, x), y) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(reach, x) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, v), y), app(app(union, i), h)), empty) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, v), y), app(app(union, i), h)) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, v), y) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(reach, v) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(union, i), h) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(union, i) APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs)) APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x)) APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x) APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs) APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(app(filter2, app(f, x)), f), x), xs) APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(filter2, app(f, x)), f), x) APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter2, app(f, x)), f) APP(app(filter, f), app(app(cons, x), xs)) -> APP(filter2, app(f, x)) APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x) APP(app(app(app(filter2, true), f), x), xs) -> APP(app(cons, x), app(app(filter, f), xs)) APP(app(app(app(filter2, true), f), x), xs) -> APP(cons, x) APP(app(app(app(filter2, true), f), x), xs) -> APP(app(filter, f), xs) APP(app(app(app(filter2, true), f), x), xs) -> APP(filter, f) APP(app(app(app(filter2, false), f), x), xs) -> APP(app(filter, f), xs) APP(app(app(app(filter2, false), f), x), xs) -> APP(filter, f) The TRS R consists of the following rules: app(app(eq, 0), 0) -> true app(app(eq, 0), app(s, x)) -> false app(app(eq, app(s, x)), 0) -> false app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y) app(app(or, true), y) -> true app(app(or, false), y) -> y app(app(union, empty), h) -> h app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h)) app(app(app(app(reach, x), y), empty), h) -> false app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) -> true app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 38 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(app(union, i), h) The TRS R consists of the following rules: app(app(eq, 0), 0) -> true app(app(eq, 0), app(s, x)) -> false app(app(eq, app(s, x)), 0) -> false app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y) app(app(or, true), y) -> true app(app(or, false), y) -> y app(app(union, empty), h) -> h app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h)) app(app(app(app(reach, x), y), empty), h) -> false app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) -> true app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(app(union, i), h) R is empty. The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) ATransformationProof (EQUIVALENT) We have applied the A-Transformation [FROCOS05] to get from an applicative problem to a standard problem. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: union1(edge(x, y, i), h) -> union1(i, h) R is empty. The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) reach(x0, x1, empty, x2) reach(x0, x1, edge(x2, x3, x4), x5) if_reach_1(true, x0, x1, edge(x2, x3, x4), x5) if_reach_1(false, x0, x1, edge(x2, x3, x4), x5) if_reach_2(true, x0, x1, edge(x2, x3, x4), x5) if_reach_2(false, x0, x1, edge(x2, x3, x4), x5) map(x0, nil) map(x0, cons(x1, x2)) filter(x0, nil) filter(x0, cons(x1, x2)) filter2(true, x0, x1, x2) filter2(false, x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) reach(x0, x1, empty, x2) reach(x0, x1, edge(x2, x3, x4), x5) if_reach_1(true, x0, x1, edge(x2, x3, x4), x5) if_reach_1(false, x0, x1, edge(x2, x3, x4), x5) if_reach_2(true, x0, x1, edge(x2, x3, x4), x5) if_reach_2(false, x0, x1, edge(x2, x3, x4), x5) map(x0, nil) map(x0, cons(x1, x2)) filter(x0, nil) filter(x0, cons(x1, x2)) filter2(true, x0, x1, x2) filter2(false, x0, x1, x2) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: union1(edge(x, y, i), h) -> union1(i, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *union1(edge(x, y, i), h) -> union1(i, h) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y) The TRS R consists of the following rules: app(app(eq, 0), 0) -> true app(app(eq, 0), app(s, x)) -> false app(app(eq, app(s, x)), 0) -> false app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y) app(app(or, true), y) -> true app(app(or, false), y) -> y app(app(union, empty), h) -> h app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h)) app(app(app(app(reach, x), y), empty), h) -> false app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) -> true app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y) R is empty. The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) ATransformationProof (EQUIVALENT) We have applied the A-Transformation [FROCOS05] to get from an applicative problem to a standard problem. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: eq1(s(x), s(y)) -> eq1(x, y) R is empty. The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) reach(x0, x1, empty, x2) reach(x0, x1, edge(x2, x3, x4), x5) if_reach_1(true, x0, x1, edge(x2, x3, x4), x5) if_reach_1(false, x0, x1, edge(x2, x3, x4), x5) if_reach_2(true, x0, x1, edge(x2, x3, x4), x5) if_reach_2(false, x0, x1, edge(x2, x3, x4), x5) map(x0, nil) map(x0, cons(x1, x2)) filter(x0, nil) filter(x0, cons(x1, x2)) filter2(true, x0, x1, x2) filter2(false, x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) reach(x0, x1, empty, x2) reach(x0, x1, edge(x2, x3, x4), x5) if_reach_1(true, x0, x1, edge(x2, x3, x4), x5) if_reach_1(false, x0, x1, edge(x2, x3, x4), x5) if_reach_2(true, x0, x1, edge(x2, x3, x4), x5) if_reach_2(false, x0, x1, edge(x2, x3, x4), x5) map(x0, nil) map(x0, cons(x1, x2)) filter(x0, nil) filter(x0, cons(x1, x2)) filter2(true, x0, x1, x2) filter2(false, x0, x1, x2) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: eq1(s(x), s(y)) -> eq1(x, y) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *eq1(s(x), s(y)) -> eq1(x, y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (24) YES ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), h) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, v), y), app(app(union, i), h)), empty) APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) The TRS R consists of the following rules: app(app(eq, 0), 0) -> true app(app(eq, 0), app(s, x)) -> false app(app(eq, app(s, x)), 0) -> false app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y) app(app(or, true), y) -> true app(app(or, false), y) -> y app(app(union, empty), h) -> h app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h)) app(app(app(app(reach, x), y), empty), h) -> false app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) -> true app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), h) APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, v), y), app(app(union, i), h)), empty) APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) The TRS R consists of the following rules: app(app(union, empty), h) -> h app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h)) app(app(eq, 0), 0) -> true app(app(eq, 0), app(s, x)) -> false app(app(eq, app(s, x)), 0) -> false app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y) The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) ATransformationProof (EQUIVALENT) We have applied the A-Transformation [FROCOS05] to get from an applicative problem to a standard problem. ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: reach1(x, y, edge(u, v, i), h) -> if_reach_11(eq(x, u), x, y, edge(u, v, i), h) if_reach_11(true, x, y, edge(u, v, i), h) -> if_reach_21(eq(y, v), x, y, edge(u, v, i), h) if_reach_21(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, h) if_reach_21(false, x, y, edge(u, v, i), h) -> reach1(v, y, union(i, h), empty) if_reach_11(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, edge(u, v, h)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) reach(x0, x1, empty, x2) reach(x0, x1, edge(x2, x3, x4), x5) if_reach_1(true, x0, x1, edge(x2, x3, x4), x5) if_reach_1(false, x0, x1, edge(x2, x3, x4), x5) if_reach_2(true, x0, x1, edge(x2, x3, x4), x5) if_reach_2(false, x0, x1, edge(x2, x3, x4), x5) map(x0, nil) map(x0, cons(x1, x2)) filter(x0, nil) filter(x0, cons(x1, x2)) filter2(true, x0, x1, x2) filter2(false, x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. or(true, x0) or(false, x0) reach(x0, x1, empty, x2) reach(x0, x1, edge(x2, x3, x4), x5) if_reach_1(true, x0, x1, edge(x2, x3, x4), x5) if_reach_1(false, x0, x1, edge(x2, x3, x4), x5) if_reach_2(true, x0, x1, edge(x2, x3, x4), x5) if_reach_2(false, x0, x1, edge(x2, x3, x4), x5) map(x0, nil) map(x0, cons(x1, x2)) filter(x0, nil) filter(x0, cons(x1, x2)) filter2(true, x0, x1, x2) filter2(false, x0, x1, x2) ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: reach1(x, y, edge(u, v, i), h) -> if_reach_11(eq(x, u), x, y, edge(u, v, i), h) if_reach_11(true, x, y, edge(u, v, i), h) -> if_reach_21(eq(y, v), x, y, edge(u, v, i), h) if_reach_21(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, h) if_reach_21(false, x, y, edge(u, v, i), h) -> reach1(v, y, union(i, h), empty) if_reach_11(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, edge(u, v, h)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. if_reach_11(true, x, y, edge(u, v, i), h) -> if_reach_21(eq(y, v), x, y, edge(u, v, i), h) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( if_reach_11_5(x_1, ..., x_5) ) = max{0, 2x_3 + x_4 + x_5 - 1} POL( if_reach_21_5(x_1, ..., x_5) ) = max{0, 2x_3 + x_4 + x_5 - 2} POL( eq_2(x_1, x_2) ) = 0 POL( 0 ) = 0 POL( true ) = 2 POL( s_1(x_1) ) = 2x_1 POL( false ) = 0 POL( reach1_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + x_4 - 1} POL( union_2(x_1, x_2) ) = x_1 + x_2 POL( empty ) = 0 POL( edge_3(x_1, ..., x_3) ) = x_3 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: reach1(x, y, edge(u, v, i), h) -> if_reach_11(eq(x, u), x, y, edge(u, v, i), h) if_reach_21(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, h) if_reach_21(false, x, y, edge(u, v, i), h) -> reach1(v, y, union(i, h), empty) if_reach_11(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, edge(u, v, h)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: if_reach_11(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, edge(u, v, h)) reach1(x, y, edge(u, v, i), h) -> if_reach_11(eq(x, u), x, y, edge(u, v, i), h) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: if_reach_11(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, edge(u, v, h)) reach1(x, y, edge(u, v, i), h) -> if_reach_11(eq(x, u), x, y, edge(u, v, i), h) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. union(empty, x0) union(edge(x0, x1, x2), x3) ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: if_reach_11(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, edge(u, v, h)) reach1(x, y, edge(u, v, i), h) -> if_reach_11(eq(x, u), x, y, edge(u, v, i), h) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (40) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *reach1(x, y, edge(u, v, i), h) -> if_reach_11(eq(x, u), x, y, edge(u, v, i), h) The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5 *if_reach_11(false, x, y, edge(u, v, i), h) -> reach1(x, y, i, edge(u, v, h)) The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3 ---------------------------------------- (41) YES ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs) APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x) APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(app(filter2, app(f, x)), f), x), xs) APP(app(app(app(filter2, true), f), x), xs) -> APP(app(filter, f), xs) APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x) APP(app(app(app(filter2, false), f), x), xs) -> APP(app(filter, f), xs) The TRS R consists of the following rules: app(app(eq, 0), 0) -> true app(app(eq, 0), app(s, x)) -> false app(app(eq, app(s, x)), 0) -> false app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y) app(app(or, true), y) -> true app(app(or, false), y) -> y app(app(union, empty), h) -> h app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h)) app(app(app(app(reach, x), y), empty), h) -> false app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h) app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h)) app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) -> true app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs) The set Q consists of the following terms: app(app(eq, 0), 0) app(app(eq, 0), app(s, x0)) app(app(eq, app(s, x0)), 0) app(app(eq, app(s, x0)), app(s, x1)) app(app(or, true), x0) app(app(or, false), x0) app(app(union, empty), x0) app(app(union, app(app(app(edge, x0), x1), x2)), x3) app(app(app(app(reach, x0), x1), empty), x2) app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5) app(app(map, x0), nil) app(app(map, x0), app(app(cons, x1), x2)) app(app(filter, x0), nil) app(app(filter, x0), app(app(cons, x1), x2)) app(app(app(app(filter2, true), x0), x1), x2) app(app(app(app(filter2, false), x0), x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x) The graph contains the following edges 1 > 1, 2 > 2 *APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x) The graph contains the following edges 1 > 1, 2 > 2 *APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs) The graph contains the following edges 1 >= 1, 2 > 2 *APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(app(filter2, app(f, x)), f), x), xs) The graph contains the following edges 2 > 2 *APP(app(app(app(filter2, true), f), x), xs) -> APP(app(filter, f), xs) The graph contains the following edges 2 >= 2 *APP(app(app(app(filter2, false), f), x), xs) -> APP(app(filter, f), xs) The graph contains the following edges 2 >= 2 ---------------------------------------- (44) YES