27.86/8.64 WORST_CASE(Omega(n^1), O(n^1)) 27.86/8.65 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 27.86/8.65 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 27.86/8.65 27.86/8.65 27.86/8.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.86/8.65 27.86/8.65 (0) CpxTRS 27.86/8.65 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 27.86/8.65 (2) CpxTRS 27.86/8.65 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 27.86/8.65 (4) CpxTRS 27.86/8.65 (5) CpxTrsMatchBoundsTAProof [FINISHED, 125 ms] 27.86/8.65 (6) BOUNDS(1, n^1) 27.86/8.65 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 27.86/8.65 (8) TRS for Loop Detection 27.86/8.65 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 27.86/8.65 (10) BEST 27.86/8.65 (11) proven lower bound 27.86/8.65 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 27.86/8.65 (13) BOUNDS(n^1, INF) 27.86/8.65 (14) TRS for Loop Detection 27.86/8.65 27.86/8.65 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (0) 27.86/8.65 Obligation: 27.86/8.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.86/8.65 27.86/8.65 27.86/8.65 The TRS R consists of the following rules: 27.86/8.65 27.86/8.65 active(U11(tt, V2)) -> mark(U12(isNat(V2))) 27.86/8.65 active(U12(tt)) -> mark(tt) 27.86/8.65 active(U21(tt)) -> mark(tt) 27.86/8.65 active(U31(tt, N)) -> mark(N) 27.86/8.65 active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) 27.86/8.65 active(U42(tt, M, N)) -> mark(s(plus(N, M))) 27.86/8.65 active(isNat(0)) -> mark(tt) 27.86/8.65 active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) 27.86/8.65 active(isNat(s(V1))) -> mark(U21(isNat(V1))) 27.86/8.65 active(plus(N, 0)) -> mark(U31(isNat(N), N)) 27.86/8.65 active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) 27.86/8.65 active(U11(X1, X2)) -> U11(active(X1), X2) 27.86/8.65 active(U12(X)) -> U12(active(X)) 27.86/8.65 active(U21(X)) -> U21(active(X)) 27.86/8.65 active(U31(X1, X2)) -> U31(active(X1), X2) 27.86/8.65 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 27.86/8.65 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 27.86/8.65 active(s(X)) -> s(active(X)) 27.86/8.65 active(plus(X1, X2)) -> plus(active(X1), X2) 27.86/8.65 active(plus(X1, X2)) -> plus(X1, active(X2)) 27.86/8.65 U11(mark(X1), X2) -> mark(U11(X1, X2)) 27.86/8.65 U12(mark(X)) -> mark(U12(X)) 27.86/8.65 U21(mark(X)) -> mark(U21(X)) 27.86/8.65 U31(mark(X1), X2) -> mark(U31(X1, X2)) 27.86/8.65 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 27.86/8.65 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 27.86/8.65 s(mark(X)) -> mark(s(X)) 27.86/8.65 plus(mark(X1), X2) -> mark(plus(X1, X2)) 27.86/8.65 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 27.86/8.65 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 27.86/8.65 proper(tt) -> ok(tt) 27.86/8.65 proper(U12(X)) -> U12(proper(X)) 27.86/8.65 proper(isNat(X)) -> isNat(proper(X)) 27.86/8.65 proper(U21(X)) -> U21(proper(X)) 27.86/8.65 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 27.86/8.65 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(s(X)) -> s(proper(X)) 27.86/8.65 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 27.86/8.65 proper(0) -> ok(0) 27.86/8.65 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 27.86/8.65 U12(ok(X)) -> ok(U12(X)) 27.86/8.65 isNat(ok(X)) -> ok(isNat(X)) 27.86/8.65 U21(ok(X)) -> ok(U21(X)) 27.86/8.65 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 27.86/8.65 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 27.86/8.65 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 27.86/8.65 s(ok(X)) -> ok(s(X)) 27.86/8.65 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 27.86/8.65 top(mark(X)) -> top(proper(X)) 27.86/8.65 top(ok(X)) -> top(active(X)) 27.86/8.65 27.86/8.65 S is empty. 27.86/8.65 Rewrite Strategy: FULL 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 27.86/8.65 The following defined symbols can occur below the 0th argument of top: proper, active 27.86/8.65 The following defined symbols can occur below the 0th argument of proper: proper, active 27.86/8.65 The following defined symbols can occur below the 0th argument of active: proper, active 27.86/8.65 27.86/8.65 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 27.86/8.65 active(U11(tt, V2)) -> mark(U12(isNat(V2))) 27.86/8.65 active(U12(tt)) -> mark(tt) 27.86/8.65 active(U21(tt)) -> mark(tt) 27.86/8.65 active(U31(tt, N)) -> mark(N) 27.86/8.65 active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) 27.86/8.65 active(U42(tt, M, N)) -> mark(s(plus(N, M))) 27.86/8.65 active(isNat(0)) -> mark(tt) 27.86/8.65 active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) 27.86/8.65 active(isNat(s(V1))) -> mark(U21(isNat(V1))) 27.86/8.65 active(plus(N, 0)) -> mark(U31(isNat(N), N)) 27.86/8.65 active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) 27.86/8.65 active(U11(X1, X2)) -> U11(active(X1), X2) 27.86/8.65 active(U12(X)) -> U12(active(X)) 27.86/8.65 active(U21(X)) -> U21(active(X)) 27.86/8.65 active(U31(X1, X2)) -> U31(active(X1), X2) 27.86/8.65 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 27.86/8.65 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 27.86/8.65 active(s(X)) -> s(active(X)) 27.86/8.65 active(plus(X1, X2)) -> plus(active(X1), X2) 27.86/8.65 active(plus(X1, X2)) -> plus(X1, active(X2)) 27.86/8.65 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 27.86/8.65 proper(U12(X)) -> U12(proper(X)) 27.86/8.65 proper(isNat(X)) -> isNat(proper(X)) 27.86/8.65 proper(U21(X)) -> U21(proper(X)) 27.86/8.65 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 27.86/8.65 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(s(X)) -> s(proper(X)) 27.86/8.65 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 27.86/8.65 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (2) 27.86/8.65 Obligation: 27.86/8.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 27.86/8.65 27.86/8.65 27.86/8.65 The TRS R consists of the following rules: 27.86/8.65 27.86/8.65 U11(mark(X1), X2) -> mark(U11(X1, X2)) 27.86/8.65 U12(mark(X)) -> mark(U12(X)) 27.86/8.65 U21(mark(X)) -> mark(U21(X)) 27.86/8.65 U31(mark(X1), X2) -> mark(U31(X1, X2)) 27.86/8.65 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 27.86/8.65 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 27.86/8.65 s(mark(X)) -> mark(s(X)) 27.86/8.65 plus(mark(X1), X2) -> mark(plus(X1, X2)) 27.86/8.65 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 27.86/8.65 proper(tt) -> ok(tt) 27.86/8.65 proper(0) -> ok(0) 27.86/8.65 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 27.86/8.65 U12(ok(X)) -> ok(U12(X)) 27.86/8.65 isNat(ok(X)) -> ok(isNat(X)) 27.86/8.65 U21(ok(X)) -> ok(U21(X)) 27.86/8.65 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 27.86/8.65 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 27.86/8.65 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 27.86/8.65 s(ok(X)) -> ok(s(X)) 27.86/8.65 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 27.86/8.65 top(mark(X)) -> top(proper(X)) 27.86/8.65 top(ok(X)) -> top(active(X)) 27.86/8.65 27.86/8.65 S is empty. 27.86/8.65 Rewrite Strategy: FULL 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 27.86/8.65 transformed relative TRS to TRS 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (4) 27.86/8.65 Obligation: 27.86/8.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 27.86/8.65 27.86/8.65 27.86/8.65 The TRS R consists of the following rules: 27.86/8.65 27.86/8.65 U11(mark(X1), X2) -> mark(U11(X1, X2)) 27.86/8.65 U12(mark(X)) -> mark(U12(X)) 27.86/8.65 U21(mark(X)) -> mark(U21(X)) 27.86/8.65 U31(mark(X1), X2) -> mark(U31(X1, X2)) 27.86/8.65 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 27.86/8.65 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 27.86/8.65 s(mark(X)) -> mark(s(X)) 27.86/8.65 plus(mark(X1), X2) -> mark(plus(X1, X2)) 27.86/8.65 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 27.86/8.65 proper(tt) -> ok(tt) 27.86/8.65 proper(0) -> ok(0) 27.86/8.65 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 27.86/8.65 U12(ok(X)) -> ok(U12(X)) 27.86/8.65 isNat(ok(X)) -> ok(isNat(X)) 27.86/8.65 U21(ok(X)) -> ok(U21(X)) 27.86/8.65 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 27.86/8.65 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 27.86/8.65 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 27.86/8.65 s(ok(X)) -> ok(s(X)) 27.86/8.65 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 27.86/8.65 top(mark(X)) -> top(proper(X)) 27.86/8.65 top(ok(X)) -> top(active(X)) 27.86/8.65 27.86/8.65 S is empty. 27.86/8.65 Rewrite Strategy: FULL 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (5) CpxTrsMatchBoundsTAProof (FINISHED) 27.86/8.65 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. 27.86/8.65 27.86/8.65 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 27.86/8.65 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] 27.86/8.65 transitions: 27.86/8.65 mark0(0) -> 0 27.86/8.65 tt0() -> 0 27.86/8.65 ok0(0) -> 0 27.86/8.65 00() -> 0 27.86/8.65 active0(0) -> 0 27.86/8.65 U110(0, 0) -> 1 27.86/8.65 U120(0) -> 2 27.86/8.65 U210(0) -> 3 27.86/8.65 U310(0, 0) -> 4 27.86/8.65 U410(0, 0, 0) -> 5 27.86/8.65 U420(0, 0, 0) -> 6 27.86/8.65 s0(0) -> 7 27.86/8.65 plus0(0, 0) -> 8 27.86/8.65 proper0(0) -> 9 27.86/8.65 isNat0(0) -> 10 27.86/8.65 top0(0) -> 11 27.86/8.65 U111(0, 0) -> 12 27.86/8.65 mark1(12) -> 1 27.86/8.65 U121(0) -> 13 27.86/8.65 mark1(13) -> 2 27.86/8.65 U211(0) -> 14 27.86/8.65 mark1(14) -> 3 27.86/8.65 U311(0, 0) -> 15 27.86/8.65 mark1(15) -> 4 27.86/8.65 U411(0, 0, 0) -> 16 27.86/8.65 mark1(16) -> 5 27.86/8.65 U421(0, 0, 0) -> 17 27.86/8.65 mark1(17) -> 6 27.86/8.65 s1(0) -> 18 27.86/8.65 mark1(18) -> 7 27.86/8.65 plus1(0, 0) -> 19 27.86/8.65 mark1(19) -> 8 27.86/8.65 tt1() -> 20 27.86/8.65 ok1(20) -> 9 27.86/8.65 01() -> 21 27.86/8.65 ok1(21) -> 9 27.86/8.65 U111(0, 0) -> 22 27.86/8.65 ok1(22) -> 1 27.86/8.65 U121(0) -> 23 27.86/8.65 ok1(23) -> 2 27.86/8.65 isNat1(0) -> 24 27.86/8.65 ok1(24) -> 10 27.86/8.65 U211(0) -> 25 27.86/8.65 ok1(25) -> 3 27.86/8.65 U311(0, 0) -> 26 27.86/8.65 ok1(26) -> 4 27.86/8.65 U411(0, 0, 0) -> 27 27.86/8.65 ok1(27) -> 5 27.86/8.65 U421(0, 0, 0) -> 28 27.86/8.65 ok1(28) -> 6 27.86/8.65 s1(0) -> 29 27.86/8.65 ok1(29) -> 7 27.86/8.65 plus1(0, 0) -> 30 27.86/8.65 ok1(30) -> 8 27.86/8.65 proper1(0) -> 31 27.86/8.65 top1(31) -> 11 27.86/8.65 active1(0) -> 32 27.86/8.65 top1(32) -> 11 27.86/8.65 mark1(12) -> 12 27.86/8.65 mark1(12) -> 22 27.86/8.65 mark1(13) -> 13 27.86/8.65 mark1(13) -> 23 27.86/8.65 mark1(14) -> 14 27.86/8.65 mark1(14) -> 25 27.86/8.65 mark1(15) -> 15 27.86/8.65 mark1(15) -> 26 27.86/8.65 mark1(16) -> 16 27.86/8.65 mark1(16) -> 27 27.86/8.65 mark1(17) -> 17 27.86/8.65 mark1(17) -> 28 27.86/8.65 mark1(18) -> 18 27.86/8.65 mark1(18) -> 29 27.86/8.65 mark1(19) -> 19 27.86/8.65 mark1(19) -> 30 27.86/8.65 ok1(20) -> 31 27.86/8.65 ok1(21) -> 31 27.86/8.65 ok1(22) -> 12 27.86/8.65 ok1(22) -> 22 27.86/8.65 ok1(23) -> 13 27.86/8.65 ok1(23) -> 23 27.86/8.65 ok1(24) -> 24 27.86/8.65 ok1(25) -> 14 27.86/8.65 ok1(25) -> 25 27.86/8.65 ok1(26) -> 15 27.86/8.65 ok1(26) -> 26 27.86/8.65 ok1(27) -> 16 27.86/8.65 ok1(27) -> 27 27.86/8.65 ok1(28) -> 17 27.86/8.65 ok1(28) -> 28 27.86/8.65 ok1(29) -> 18 27.86/8.65 ok1(29) -> 29 27.86/8.65 ok1(30) -> 19 27.86/8.65 ok1(30) -> 30 27.86/8.65 active2(20) -> 33 27.86/8.65 top2(33) -> 11 27.86/8.65 active2(21) -> 33 27.86/8.65 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (6) 27.86/8.65 BOUNDS(1, n^1) 27.86/8.65 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 27.86/8.65 Transformed a relative TRS into a decreasing-loop problem. 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (8) 27.86/8.65 Obligation: 27.86/8.65 Analyzing the following TRS for decreasing loops: 27.86/8.65 27.86/8.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.86/8.65 27.86/8.65 27.86/8.65 The TRS R consists of the following rules: 27.86/8.65 27.86/8.65 active(U11(tt, V2)) -> mark(U12(isNat(V2))) 27.86/8.65 active(U12(tt)) -> mark(tt) 27.86/8.65 active(U21(tt)) -> mark(tt) 27.86/8.65 active(U31(tt, N)) -> mark(N) 27.86/8.65 active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) 27.86/8.65 active(U42(tt, M, N)) -> mark(s(plus(N, M))) 27.86/8.65 active(isNat(0)) -> mark(tt) 27.86/8.65 active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) 27.86/8.65 active(isNat(s(V1))) -> mark(U21(isNat(V1))) 27.86/8.65 active(plus(N, 0)) -> mark(U31(isNat(N), N)) 27.86/8.65 active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) 27.86/8.65 active(U11(X1, X2)) -> U11(active(X1), X2) 27.86/8.65 active(U12(X)) -> U12(active(X)) 27.86/8.65 active(U21(X)) -> U21(active(X)) 27.86/8.65 active(U31(X1, X2)) -> U31(active(X1), X2) 27.86/8.65 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 27.86/8.65 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 27.86/8.65 active(s(X)) -> s(active(X)) 27.86/8.65 active(plus(X1, X2)) -> plus(active(X1), X2) 27.86/8.65 active(plus(X1, X2)) -> plus(X1, active(X2)) 27.86/8.65 U11(mark(X1), X2) -> mark(U11(X1, X2)) 27.86/8.65 U12(mark(X)) -> mark(U12(X)) 27.86/8.65 U21(mark(X)) -> mark(U21(X)) 27.86/8.65 U31(mark(X1), X2) -> mark(U31(X1, X2)) 27.86/8.65 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 27.86/8.65 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 27.86/8.65 s(mark(X)) -> mark(s(X)) 27.86/8.65 plus(mark(X1), X2) -> mark(plus(X1, X2)) 27.86/8.65 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 27.86/8.65 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 27.86/8.65 proper(tt) -> ok(tt) 27.86/8.65 proper(U12(X)) -> U12(proper(X)) 27.86/8.65 proper(isNat(X)) -> isNat(proper(X)) 27.86/8.65 proper(U21(X)) -> U21(proper(X)) 27.86/8.65 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 27.86/8.65 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(s(X)) -> s(proper(X)) 27.86/8.65 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 27.86/8.65 proper(0) -> ok(0) 27.86/8.65 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 27.86/8.65 U12(ok(X)) -> ok(U12(X)) 27.86/8.65 isNat(ok(X)) -> ok(isNat(X)) 27.86/8.65 U21(ok(X)) -> ok(U21(X)) 27.86/8.65 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 27.86/8.65 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 27.86/8.65 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 27.86/8.65 s(ok(X)) -> ok(s(X)) 27.86/8.65 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 27.86/8.65 top(mark(X)) -> top(proper(X)) 27.86/8.65 top(ok(X)) -> top(active(X)) 27.86/8.65 27.86/8.65 S is empty. 27.86/8.65 Rewrite Strategy: FULL 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (9) DecreasingLoopProof (LOWER BOUND(ID)) 27.86/8.65 The following loop(s) give(s) rise to the lower bound Omega(n^1): 27.86/8.65 27.86/8.65 The rewrite sequence 27.86/8.65 27.86/8.65 plus(X1, mark(X2)) ->^+ mark(plus(X1, X2)) 27.86/8.65 27.86/8.65 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 27.86/8.65 27.86/8.65 The pumping substitution is [X2 / mark(X2)]. 27.86/8.65 27.86/8.65 The result substitution is [ ]. 27.86/8.65 27.86/8.65 27.86/8.65 27.86/8.65 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (10) 27.86/8.65 Complex Obligation (BEST) 27.86/8.65 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (11) 27.86/8.65 Obligation: 27.86/8.65 Proved the lower bound n^1 for the following obligation: 27.86/8.65 27.86/8.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.86/8.65 27.86/8.65 27.86/8.65 The TRS R consists of the following rules: 27.86/8.65 27.86/8.65 active(U11(tt, V2)) -> mark(U12(isNat(V2))) 27.86/8.65 active(U12(tt)) -> mark(tt) 27.86/8.65 active(U21(tt)) -> mark(tt) 27.86/8.65 active(U31(tt, N)) -> mark(N) 27.86/8.65 active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) 27.86/8.65 active(U42(tt, M, N)) -> mark(s(plus(N, M))) 27.86/8.65 active(isNat(0)) -> mark(tt) 27.86/8.65 active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) 27.86/8.65 active(isNat(s(V1))) -> mark(U21(isNat(V1))) 27.86/8.65 active(plus(N, 0)) -> mark(U31(isNat(N), N)) 27.86/8.65 active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) 27.86/8.65 active(U11(X1, X2)) -> U11(active(X1), X2) 27.86/8.65 active(U12(X)) -> U12(active(X)) 27.86/8.65 active(U21(X)) -> U21(active(X)) 27.86/8.65 active(U31(X1, X2)) -> U31(active(X1), X2) 27.86/8.65 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 27.86/8.65 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 27.86/8.65 active(s(X)) -> s(active(X)) 27.86/8.65 active(plus(X1, X2)) -> plus(active(X1), X2) 27.86/8.65 active(plus(X1, X2)) -> plus(X1, active(X2)) 27.86/8.65 U11(mark(X1), X2) -> mark(U11(X1, X2)) 27.86/8.65 U12(mark(X)) -> mark(U12(X)) 27.86/8.65 U21(mark(X)) -> mark(U21(X)) 27.86/8.65 U31(mark(X1), X2) -> mark(U31(X1, X2)) 27.86/8.65 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 27.86/8.65 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 27.86/8.65 s(mark(X)) -> mark(s(X)) 27.86/8.65 plus(mark(X1), X2) -> mark(plus(X1, X2)) 27.86/8.65 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 27.86/8.65 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 27.86/8.65 proper(tt) -> ok(tt) 27.86/8.65 proper(U12(X)) -> U12(proper(X)) 27.86/8.65 proper(isNat(X)) -> isNat(proper(X)) 27.86/8.65 proper(U21(X)) -> U21(proper(X)) 27.86/8.65 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 27.86/8.65 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(s(X)) -> s(proper(X)) 27.86/8.65 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 27.86/8.65 proper(0) -> ok(0) 27.86/8.65 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 27.86/8.65 U12(ok(X)) -> ok(U12(X)) 27.86/8.65 isNat(ok(X)) -> ok(isNat(X)) 27.86/8.65 U21(ok(X)) -> ok(U21(X)) 27.86/8.65 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 27.86/8.65 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 27.86/8.65 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 27.86/8.65 s(ok(X)) -> ok(s(X)) 27.86/8.65 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 27.86/8.65 top(mark(X)) -> top(proper(X)) 27.86/8.65 top(ok(X)) -> top(active(X)) 27.86/8.65 27.86/8.65 S is empty. 27.86/8.65 Rewrite Strategy: FULL 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (12) LowerBoundPropagationProof (FINISHED) 27.86/8.65 Propagated lower bound. 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (13) 27.86/8.65 BOUNDS(n^1, INF) 27.86/8.65 27.86/8.65 ---------------------------------------- 27.86/8.65 27.86/8.65 (14) 27.86/8.65 Obligation: 27.86/8.65 Analyzing the following TRS for decreasing loops: 27.86/8.65 27.86/8.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.86/8.65 27.86/8.65 27.86/8.65 The TRS R consists of the following rules: 27.86/8.65 27.86/8.65 active(U11(tt, V2)) -> mark(U12(isNat(V2))) 27.86/8.65 active(U12(tt)) -> mark(tt) 27.86/8.65 active(U21(tt)) -> mark(tt) 27.86/8.65 active(U31(tt, N)) -> mark(N) 27.86/8.65 active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) 27.86/8.65 active(U42(tt, M, N)) -> mark(s(plus(N, M))) 27.86/8.65 active(isNat(0)) -> mark(tt) 27.86/8.65 active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) 27.86/8.65 active(isNat(s(V1))) -> mark(U21(isNat(V1))) 27.86/8.65 active(plus(N, 0)) -> mark(U31(isNat(N), N)) 27.86/8.65 active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) 27.86/8.65 active(U11(X1, X2)) -> U11(active(X1), X2) 27.86/8.65 active(U12(X)) -> U12(active(X)) 27.86/8.65 active(U21(X)) -> U21(active(X)) 27.86/8.65 active(U31(X1, X2)) -> U31(active(X1), X2) 27.86/8.65 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 27.86/8.65 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 27.86/8.65 active(s(X)) -> s(active(X)) 27.86/8.65 active(plus(X1, X2)) -> plus(active(X1), X2) 27.86/8.65 active(plus(X1, X2)) -> plus(X1, active(X2)) 27.86/8.65 U11(mark(X1), X2) -> mark(U11(X1, X2)) 27.86/8.65 U12(mark(X)) -> mark(U12(X)) 27.86/8.65 U21(mark(X)) -> mark(U21(X)) 27.86/8.65 U31(mark(X1), X2) -> mark(U31(X1, X2)) 27.86/8.65 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 27.86/8.65 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 27.86/8.65 s(mark(X)) -> mark(s(X)) 27.86/8.65 plus(mark(X1), X2) -> mark(plus(X1, X2)) 27.86/8.65 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 27.86/8.65 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 27.86/8.65 proper(tt) -> ok(tt) 27.86/8.65 proper(U12(X)) -> U12(proper(X)) 27.86/8.65 proper(isNat(X)) -> isNat(proper(X)) 27.86/8.65 proper(U21(X)) -> U21(proper(X)) 27.86/8.65 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 27.86/8.65 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 27.86/8.65 proper(s(X)) -> s(proper(X)) 27.86/8.65 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 27.86/8.65 proper(0) -> ok(0) 27.86/8.65 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 27.86/8.65 U12(ok(X)) -> ok(U12(X)) 27.86/8.65 isNat(ok(X)) -> ok(isNat(X)) 27.86/8.65 U21(ok(X)) -> ok(U21(X)) 27.86/8.65 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 27.86/8.65 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 27.86/8.65 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 27.86/8.65 s(ok(X)) -> ok(s(X)) 27.86/8.65 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 27.86/8.65 top(mark(X)) -> top(proper(X)) 27.86/8.65 top(ok(X)) -> top(active(X)) 27.86/8.65 27.86/8.65 S is empty. 27.86/8.65 Rewrite Strategy: FULL 28.55/8.91 EOF