/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) Overlay + Local Confluence [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof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, 15 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 ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(member, w), null) -> false app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))) 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(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(member, w), null) -> false app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))) The set Q consists of the following terms: app(app(lt, app(s, x0)), app(s, x1)) app(app(lt, 0), app(s, x0)) app(app(lt, x0), 0) app(app(eq, x0), x0) app(app(eq, app(s, x0)), 0) app(app(eq, 0), app(s, x0)) app(app(member, x0), null) app(app(member, x0), app(app(app(fork, x1), x2), x3)) ---------------------------------------- (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(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y) APP(app(lt, app(s, x)), app(s, y)) -> APP(lt, x) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x)) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(if, app(app(lt, w), y)) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(lt, w) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(eq, w), y)), true) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(if, app(app(eq, w), y)) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(eq, w), y) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(eq, w) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z) The TRS R consists of the following rules: app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(member, w), null) -> false app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))) The set Q consists of the following terms: app(app(lt, app(s, x0)), app(s, x1)) app(app(lt, 0), app(s, x0)) app(app(lt, x0), 0) app(app(eq, x0), x0) app(app(eq, app(s, x0)), 0) app(app(eq, 0), app(s, x0)) app(app(member, x0), null) app(app(member, x0), app(app(app(fork, x1), x2), x3)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 11 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y) The TRS R consists of the following rules: app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(member, w), null) -> false app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))) The set Q consists of the following terms: app(app(lt, app(s, x0)), app(s, x1)) app(app(lt, 0), app(s, x0)) app(app(lt, x0), 0) app(app(eq, x0), x0) app(app(eq, app(s, x0)), 0) app(app(eq, 0), app(s, x0)) app(app(member, x0), null) app(app(member, x0), app(app(app(fork, x1), x2), x3)) 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(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y) R is empty. The set Q consists of the following terms: app(app(lt, app(s, x0)), app(s, x1)) app(app(lt, 0), app(s, x0)) app(app(lt, x0), 0) app(app(eq, x0), x0) app(app(eq, app(s, x0)), 0) app(app(eq, 0), app(s, x0)) app(app(member, x0), null) app(app(member, x0), app(app(app(fork, x1), x2), x3)) 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: lt1(s(x), s(y)) -> lt1(x, y) R is empty. The set Q consists of the following terms: lt(s(x0), s(x1)) lt(0, s(x0)) lt(x0, 0) eq(x0, x0) eq(s(x0), 0) eq(0, s(x0)) member(x0, null) member(x0, fork(x1, x2, x3)) 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]. lt(s(x0), s(x1)) lt(0, s(x0)) lt(x0, 0) eq(x0, x0) eq(s(x0), 0) eq(0, s(x0)) member(x0, null) member(x0, fork(x1, x2, x3)) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: lt1(s(x), s(y)) -> lt1(x, y) 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: *lt1(s(x), s(y)) -> lt1(x, y) 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(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x) The TRS R consists of the following rules: app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(member, w), null) -> false app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))) The set Q consists of the following terms: app(app(lt, app(s, x0)), app(s, x1)) app(app(lt, 0), app(s, x0)) app(app(lt, x0), 0) app(app(eq, x0), x0) app(app(eq, app(s, x0)), 0) app(app(eq, 0), app(s, x0)) app(app(member, x0), null) app(app(member, x0), app(app(app(fork, x1), x2), x3)) 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(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z) APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x) R is empty. The set Q consists of the following terms: app(app(lt, app(s, x0)), app(s, x1)) app(app(lt, 0), app(s, x0)) app(app(lt, x0), 0) app(app(eq, x0), x0) app(app(eq, app(s, x0)), 0) app(app(eq, 0), app(s, x0)) app(app(member, x0), null) app(app(member, x0), app(app(app(fork, x1), x2), x3)) 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: member1(w, fork(x, y, z)) -> member1(w, z) member1(w, fork(x, y, z)) -> member1(w, x) R is empty. The set Q consists of the following terms: lt(s(x0), s(x1)) lt(0, s(x0)) lt(x0, 0) eq(x0, x0) eq(s(x0), 0) eq(0, s(x0)) member(x0, null) member(x0, fork(x1, x2, x3)) 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]. lt(s(x0), s(x1)) lt(0, s(x0)) lt(x0, 0) eq(x0, x0) eq(s(x0), 0) eq(0, s(x0)) member(x0, null) member(x0, fork(x1, x2, x3)) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: member1(w, fork(x, y, z)) -> member1(w, z) member1(w, fork(x, y, z)) -> member1(w, x) 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: *member1(w, fork(x, y, z)) -> member1(w, z) The graph contains the following edges 1 >= 1, 2 > 2 *member1(w, fork(x, y, z)) -> member1(w, x) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (24) YES