/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 9 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 158 ms] (6) BOUNDS(1, n^1) (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (8) TRS for Loop Detection (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (10) BEST (11) proven lower bound (12) LowerBoundPropagationProof [FINISHED, 0 ms] (13) BOUNDS(n^1, INF) (14) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(0) -> ok(0) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNat(ok(X)) -> ok(isNat(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(U12(X)) -> U12(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNat(ok(X)) -> ok(isNat(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNat(ok(X)) -> ok(isNat(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (5) CpxTrsMatchBoundsTAProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] transitions: mark0(0) -> 0 tt0() -> 0 ok0(0) -> 0 00() -> 0 active0(0) -> 0 U110(0, 0) -> 1 U120(0) -> 2 U210(0) -> 3 U310(0, 0) -> 4 U410(0, 0, 0) -> 5 U420(0, 0, 0) -> 6 s0(0) -> 7 plus0(0, 0) -> 8 proper0(0) -> 9 isNat0(0) -> 10 top0(0) -> 11 U111(0, 0) -> 12 mark1(12) -> 1 U121(0) -> 13 mark1(13) -> 2 U211(0) -> 14 mark1(14) -> 3 U311(0, 0) -> 15 mark1(15) -> 4 U411(0, 0, 0) -> 16 mark1(16) -> 5 U421(0, 0, 0) -> 17 mark1(17) -> 6 s1(0) -> 18 mark1(18) -> 7 plus1(0, 0) -> 19 mark1(19) -> 8 tt1() -> 20 ok1(20) -> 9 01() -> 21 ok1(21) -> 9 U111(0, 0) -> 22 ok1(22) -> 1 U121(0) -> 23 ok1(23) -> 2 isNat1(0) -> 24 ok1(24) -> 10 U211(0) -> 25 ok1(25) -> 3 U311(0, 0) -> 26 ok1(26) -> 4 U411(0, 0, 0) -> 27 ok1(27) -> 5 U421(0, 0, 0) -> 28 ok1(28) -> 6 s1(0) -> 29 ok1(29) -> 7 plus1(0, 0) -> 30 ok1(30) -> 8 proper1(0) -> 31 top1(31) -> 11 active1(0) -> 32 top1(32) -> 11 mark1(12) -> 12 mark1(12) -> 22 mark1(13) -> 13 mark1(13) -> 23 mark1(14) -> 14 mark1(14) -> 25 mark1(15) -> 15 mark1(15) -> 26 mark1(16) -> 16 mark1(16) -> 27 mark1(17) -> 17 mark1(17) -> 28 mark1(18) -> 18 mark1(18) -> 29 mark1(19) -> 19 mark1(19) -> 30 ok1(20) -> 31 ok1(21) -> 31 ok1(22) -> 12 ok1(22) -> 22 ok1(23) -> 13 ok1(23) -> 23 ok1(24) -> 24 ok1(25) -> 14 ok1(25) -> 25 ok1(26) -> 15 ok1(26) -> 26 ok1(27) -> 16 ok1(27) -> 27 ok1(28) -> 17 ok1(28) -> 28 ok1(29) -> 18 ok1(29) -> 29 ok1(30) -> 19 ok1(30) -> 30 active2(20) -> 33 top2(33) -> 11 active2(21) -> 33 ---------------------------------------- (6) BOUNDS(1, n^1) ---------------------------------------- (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(0) -> ok(0) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNat(ok(X)) -> ok(isNat(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence plus(X1, mark(X2)) ->^+ mark(plus(X1, X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X2 / mark(X2)]. The result substitution is [ ]. ---------------------------------------- (10) Complex Obligation (BEST) ---------------------------------------- (11) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(0) -> ok(0) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNat(ok(X)) -> ok(isNat(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (12) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (13) BOUNDS(n^1, INF) ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(0) -> ok(0) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNat(ok(X)) -> ok(isNat(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL