24.08/8.28 WORST_CASE(Omega(n^1), O(n^1)) 24.08/8.29 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.08/8.29 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.08/8.29 24.08/8.29 24.08/8.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.08/8.29 24.08/8.29 (0) CpxTRS 24.08/8.29 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 6 ms] 24.08/8.29 (2) CpxTRS 24.08/8.29 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.08/8.29 (4) CpxTRS 24.08/8.29 (5) CpxTrsMatchBoundsTAProof [FINISHED, 58 ms] 24.08/8.29 (6) BOUNDS(1, n^1) 24.08/8.29 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 24.08/8.29 (8) TRS for Loop Detection 24.08/8.29 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 24.08/8.29 (10) BEST 24.08/8.29 (11) proven lower bound 24.08/8.29 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 24.08/8.29 (13) BOUNDS(n^1, INF) 24.08/8.29 (14) TRS for Loop Detection 24.08/8.29 24.08/8.29 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (0) 24.08/8.29 Obligation: 24.08/8.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.08/8.29 24.08/8.29 24.08/8.29 The TRS R consists of the following rules: 24.08/8.29 24.08/8.29 active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) 24.08/8.29 active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) 24.08/8.29 active(add(0, X)) -> mark(X) 24.08/8.29 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.08/8.29 active(sel(0, cons(X, XS))) -> mark(X) 24.08/8.29 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 24.08/8.29 active(fib(X)) -> fib(active(X)) 24.08/8.29 active(sel(X1, X2)) -> sel(active(X1), X2) 24.08/8.29 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.08/8.29 active(fib1(X1, X2)) -> fib1(active(X1), X2) 24.08/8.29 active(fib1(X1, X2)) -> fib1(X1, active(X2)) 24.08/8.29 active(s(X)) -> s(active(X)) 24.08/8.29 active(cons(X1, X2)) -> cons(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(X1, active(X2)) 24.08/8.29 fib(mark(X)) -> mark(fib(X)) 24.08/8.29 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.08/8.29 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.08/8.29 fib1(mark(X1), X2) -> mark(fib1(X1, X2)) 24.08/8.29 fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) 24.08/8.29 s(mark(X)) -> mark(s(X)) 24.08/8.29 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.08/8.29 add(mark(X1), X2) -> mark(add(X1, X2)) 24.08/8.29 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.08/8.29 proper(fib(X)) -> fib(proper(X)) 24.08/8.29 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.08/8.29 proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) 24.08/8.29 proper(s(X)) -> s(proper(X)) 24.08/8.29 proper(0) -> ok(0) 24.08/8.29 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.08/8.29 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.08/8.29 fib(ok(X)) -> ok(fib(X)) 24.08/8.29 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.08/8.29 fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) 24.08/8.29 s(ok(X)) -> ok(s(X)) 24.08/8.29 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.08/8.29 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.08/8.29 top(mark(X)) -> top(proper(X)) 24.08/8.29 top(ok(X)) -> top(active(X)) 24.08/8.29 24.08/8.29 S is empty. 24.08/8.29 Rewrite Strategy: FULL 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.08/8.29 The following defined symbols can occur below the 0th argument of top: proper, active 24.08/8.29 The following defined symbols can occur below the 0th argument of proper: proper, active 24.08/8.29 The following defined symbols can occur below the 0th argument of active: proper, active 24.08/8.29 24.08/8.29 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.08/8.29 active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) 24.08/8.29 active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) 24.08/8.29 active(add(0, X)) -> mark(X) 24.08/8.29 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.08/8.29 active(sel(0, cons(X, XS))) -> mark(X) 24.08/8.29 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 24.08/8.29 active(fib(X)) -> fib(active(X)) 24.08/8.29 active(sel(X1, X2)) -> sel(active(X1), X2) 24.08/8.29 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.08/8.29 active(fib1(X1, X2)) -> fib1(active(X1), X2) 24.08/8.29 active(fib1(X1, X2)) -> fib1(X1, active(X2)) 24.08/8.29 active(s(X)) -> s(active(X)) 24.08/8.29 active(cons(X1, X2)) -> cons(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(X1, active(X2)) 24.08/8.29 proper(fib(X)) -> fib(proper(X)) 24.08/8.29 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.08/8.29 proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) 24.08/8.29 proper(s(X)) -> s(proper(X)) 24.08/8.29 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.08/8.29 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.08/8.29 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (2) 24.08/8.29 Obligation: 24.08/8.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.08/8.29 24.08/8.29 24.08/8.29 The TRS R consists of the following rules: 24.08/8.29 24.08/8.29 fib(mark(X)) -> mark(fib(X)) 24.08/8.29 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.08/8.29 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.08/8.29 fib1(mark(X1), X2) -> mark(fib1(X1, X2)) 24.08/8.29 fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) 24.08/8.29 s(mark(X)) -> mark(s(X)) 24.08/8.29 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.08/8.29 add(mark(X1), X2) -> mark(add(X1, X2)) 24.08/8.29 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.08/8.29 proper(0) -> ok(0) 24.08/8.29 fib(ok(X)) -> ok(fib(X)) 24.08/8.29 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.08/8.29 fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) 24.08/8.29 s(ok(X)) -> ok(s(X)) 24.08/8.29 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.08/8.29 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.08/8.29 top(mark(X)) -> top(proper(X)) 24.08/8.29 top(ok(X)) -> top(active(X)) 24.08/8.29 24.08/8.29 S is empty. 24.08/8.29 Rewrite Strategy: FULL 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.08/8.29 transformed relative TRS to TRS 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (4) 24.08/8.29 Obligation: 24.08/8.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.08/8.29 24.08/8.29 24.08/8.29 The TRS R consists of the following rules: 24.08/8.29 24.08/8.29 fib(mark(X)) -> mark(fib(X)) 24.08/8.29 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.08/8.29 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.08/8.29 fib1(mark(X1), X2) -> mark(fib1(X1, X2)) 24.08/8.29 fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) 24.08/8.29 s(mark(X)) -> mark(s(X)) 24.08/8.29 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.08/8.29 add(mark(X1), X2) -> mark(add(X1, X2)) 24.08/8.29 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.08/8.29 proper(0) -> ok(0) 24.08/8.29 fib(ok(X)) -> ok(fib(X)) 24.08/8.29 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.08/8.29 fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) 24.08/8.29 s(ok(X)) -> ok(s(X)) 24.08/8.29 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.08/8.29 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.08/8.29 top(mark(X)) -> top(proper(X)) 24.08/8.29 top(ok(X)) -> top(active(X)) 24.08/8.29 24.08/8.29 S is empty. 24.08/8.29 Rewrite Strategy: FULL 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.08/8.29 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. 24.08/8.29 24.08/8.29 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.08/8.29 final states : [1, 2, 3, 4, 5, 6, 7, 8] 24.08/8.29 transitions: 24.08/8.29 mark0(0) -> 0 24.08/8.29 00() -> 0 24.08/8.29 ok0(0) -> 0 24.08/8.29 active0(0) -> 0 24.08/8.29 fib0(0) -> 1 24.08/8.29 sel0(0, 0) -> 2 24.08/8.29 fib10(0, 0) -> 3 24.08/8.29 s0(0) -> 4 24.08/8.29 cons0(0, 0) -> 5 24.08/8.29 add0(0, 0) -> 6 24.08/8.29 proper0(0) -> 7 24.08/8.29 top0(0) -> 8 24.08/8.29 fib1(0) -> 9 24.08/8.29 mark1(9) -> 1 24.08/8.29 sel1(0, 0) -> 10 24.08/8.29 mark1(10) -> 2 24.08/8.29 fib11(0, 0) -> 11 24.08/8.29 mark1(11) -> 3 24.08/8.29 s1(0) -> 12 24.08/8.29 mark1(12) -> 4 24.08/8.29 cons1(0, 0) -> 13 24.08/8.29 mark1(13) -> 5 24.08/8.29 add1(0, 0) -> 14 24.08/8.29 mark1(14) -> 6 24.08/8.29 01() -> 15 24.08/8.29 ok1(15) -> 7 24.08/8.29 fib1(0) -> 16 24.08/8.29 ok1(16) -> 1 24.08/8.29 sel1(0, 0) -> 17 24.08/8.29 ok1(17) -> 2 24.08/8.29 fib11(0, 0) -> 18 24.08/8.29 ok1(18) -> 3 24.08/8.29 s1(0) -> 19 24.08/8.29 ok1(19) -> 4 24.08/8.29 cons1(0, 0) -> 20 24.08/8.29 ok1(20) -> 5 24.08/8.29 add1(0, 0) -> 21 24.08/8.29 ok1(21) -> 6 24.08/8.29 proper1(0) -> 22 24.08/8.29 top1(22) -> 8 24.08/8.29 active1(0) -> 23 24.08/8.29 top1(23) -> 8 24.08/8.29 mark1(9) -> 9 24.08/8.29 mark1(9) -> 16 24.08/8.29 mark1(10) -> 10 24.08/8.29 mark1(10) -> 17 24.08/8.29 mark1(11) -> 11 24.08/8.29 mark1(11) -> 18 24.08/8.29 mark1(12) -> 12 24.08/8.29 mark1(12) -> 19 24.08/8.29 mark1(13) -> 13 24.08/8.29 mark1(13) -> 20 24.08/8.29 mark1(14) -> 14 24.08/8.29 mark1(14) -> 21 24.08/8.29 ok1(15) -> 22 24.08/8.29 ok1(16) -> 9 24.08/8.29 ok1(16) -> 16 24.08/8.29 ok1(17) -> 10 24.08/8.29 ok1(17) -> 17 24.08/8.29 ok1(18) -> 11 24.08/8.29 ok1(18) -> 18 24.08/8.29 ok1(19) -> 12 24.08/8.29 ok1(19) -> 19 24.08/8.29 ok1(20) -> 13 24.08/8.29 ok1(20) -> 20 24.08/8.29 ok1(21) -> 14 24.08/8.29 ok1(21) -> 21 24.08/8.29 active2(15) -> 24 24.08/8.29 top2(24) -> 8 24.08/8.29 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (6) 24.08/8.29 BOUNDS(1, n^1) 24.08/8.29 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 24.08/8.29 Transformed a relative TRS into a decreasing-loop problem. 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (8) 24.08/8.29 Obligation: 24.08/8.29 Analyzing the following TRS for decreasing loops: 24.08/8.29 24.08/8.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.08/8.29 24.08/8.29 24.08/8.29 The TRS R consists of the following rules: 24.08/8.29 24.08/8.29 active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) 24.08/8.29 active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) 24.08/8.29 active(add(0, X)) -> mark(X) 24.08/8.29 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.08/8.29 active(sel(0, cons(X, XS))) -> mark(X) 24.08/8.29 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 24.08/8.29 active(fib(X)) -> fib(active(X)) 24.08/8.29 active(sel(X1, X2)) -> sel(active(X1), X2) 24.08/8.29 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.08/8.29 active(fib1(X1, X2)) -> fib1(active(X1), X2) 24.08/8.29 active(fib1(X1, X2)) -> fib1(X1, active(X2)) 24.08/8.29 active(s(X)) -> s(active(X)) 24.08/8.29 active(cons(X1, X2)) -> cons(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(X1, active(X2)) 24.08/8.29 fib(mark(X)) -> mark(fib(X)) 24.08/8.29 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.08/8.29 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.08/8.29 fib1(mark(X1), X2) -> mark(fib1(X1, X2)) 24.08/8.29 fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) 24.08/8.29 s(mark(X)) -> mark(s(X)) 24.08/8.29 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.08/8.29 add(mark(X1), X2) -> mark(add(X1, X2)) 24.08/8.29 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.08/8.29 proper(fib(X)) -> fib(proper(X)) 24.08/8.29 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.08/8.29 proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) 24.08/8.29 proper(s(X)) -> s(proper(X)) 24.08/8.29 proper(0) -> ok(0) 24.08/8.29 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.08/8.29 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.08/8.29 fib(ok(X)) -> ok(fib(X)) 24.08/8.29 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.08/8.29 fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) 24.08/8.29 s(ok(X)) -> ok(s(X)) 24.08/8.29 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.08/8.29 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.08/8.29 top(mark(X)) -> top(proper(X)) 24.08/8.29 top(ok(X)) -> top(active(X)) 24.08/8.29 24.08/8.29 S is empty. 24.08/8.29 Rewrite Strategy: FULL 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (9) DecreasingLoopProof (LOWER BOUND(ID)) 24.08/8.29 The following loop(s) give(s) rise to the lower bound Omega(n^1): 24.08/8.29 24.08/8.29 The rewrite sequence 24.08/8.29 24.08/8.29 fib1(ok(X1), ok(X2)) ->^+ ok(fib1(X1, X2)) 24.08/8.29 24.08/8.29 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 24.08/8.29 24.08/8.29 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 24.08/8.29 24.08/8.29 The result substitution is [ ]. 24.08/8.29 24.08/8.29 24.08/8.29 24.08/8.29 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (10) 24.08/8.29 Complex Obligation (BEST) 24.08/8.29 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (11) 24.08/8.29 Obligation: 24.08/8.29 Proved the lower bound n^1 for the following obligation: 24.08/8.29 24.08/8.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.08/8.29 24.08/8.29 24.08/8.29 The TRS R consists of the following rules: 24.08/8.29 24.08/8.29 active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) 24.08/8.29 active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) 24.08/8.29 active(add(0, X)) -> mark(X) 24.08/8.29 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.08/8.29 active(sel(0, cons(X, XS))) -> mark(X) 24.08/8.29 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 24.08/8.29 active(fib(X)) -> fib(active(X)) 24.08/8.29 active(sel(X1, X2)) -> sel(active(X1), X2) 24.08/8.29 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.08/8.29 active(fib1(X1, X2)) -> fib1(active(X1), X2) 24.08/8.29 active(fib1(X1, X2)) -> fib1(X1, active(X2)) 24.08/8.29 active(s(X)) -> s(active(X)) 24.08/8.29 active(cons(X1, X2)) -> cons(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(X1, active(X2)) 24.08/8.29 fib(mark(X)) -> mark(fib(X)) 24.08/8.29 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.08/8.29 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.08/8.29 fib1(mark(X1), X2) -> mark(fib1(X1, X2)) 24.08/8.29 fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) 24.08/8.29 s(mark(X)) -> mark(s(X)) 24.08/8.29 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.08/8.29 add(mark(X1), X2) -> mark(add(X1, X2)) 24.08/8.29 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.08/8.29 proper(fib(X)) -> fib(proper(X)) 24.08/8.29 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.08/8.29 proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) 24.08/8.29 proper(s(X)) -> s(proper(X)) 24.08/8.29 proper(0) -> ok(0) 24.08/8.29 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.08/8.29 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.08/8.29 fib(ok(X)) -> ok(fib(X)) 24.08/8.29 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.08/8.29 fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) 24.08/8.29 s(ok(X)) -> ok(s(X)) 24.08/8.29 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.08/8.29 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.08/8.29 top(mark(X)) -> top(proper(X)) 24.08/8.29 top(ok(X)) -> top(active(X)) 24.08/8.29 24.08/8.29 S is empty. 24.08/8.29 Rewrite Strategy: FULL 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (12) LowerBoundPropagationProof (FINISHED) 24.08/8.29 Propagated lower bound. 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (13) 24.08/8.29 BOUNDS(n^1, INF) 24.08/8.29 24.08/8.29 ---------------------------------------- 24.08/8.29 24.08/8.29 (14) 24.08/8.29 Obligation: 24.08/8.29 Analyzing the following TRS for decreasing loops: 24.08/8.29 24.08/8.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.08/8.29 24.08/8.29 24.08/8.29 The TRS R consists of the following rules: 24.08/8.29 24.08/8.29 active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) 24.08/8.29 active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) 24.08/8.29 active(add(0, X)) -> mark(X) 24.08/8.29 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.08/8.29 active(sel(0, cons(X, XS))) -> mark(X) 24.08/8.29 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 24.08/8.29 active(fib(X)) -> fib(active(X)) 24.08/8.29 active(sel(X1, X2)) -> sel(active(X1), X2) 24.08/8.29 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.08/8.29 active(fib1(X1, X2)) -> fib1(active(X1), X2) 24.08/8.29 active(fib1(X1, X2)) -> fib1(X1, active(X2)) 24.08/8.29 active(s(X)) -> s(active(X)) 24.08/8.29 active(cons(X1, X2)) -> cons(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(active(X1), X2) 24.08/8.29 active(add(X1, X2)) -> add(X1, active(X2)) 24.08/8.29 fib(mark(X)) -> mark(fib(X)) 24.08/8.29 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.08/8.29 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.08/8.29 fib1(mark(X1), X2) -> mark(fib1(X1, X2)) 24.08/8.29 fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) 24.08/8.29 s(mark(X)) -> mark(s(X)) 24.08/8.29 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.08/8.29 add(mark(X1), X2) -> mark(add(X1, X2)) 24.08/8.29 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.08/8.29 proper(fib(X)) -> fib(proper(X)) 24.08/8.29 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.08/8.29 proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) 24.08/8.29 proper(s(X)) -> s(proper(X)) 24.08/8.29 proper(0) -> ok(0) 24.08/8.29 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.08/8.29 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.08/8.29 fib(ok(X)) -> ok(fib(X)) 24.08/8.29 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.08/8.29 fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) 24.08/8.29 s(ok(X)) -> ok(s(X)) 24.08/8.29 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.08/8.29 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.08/8.29 top(mark(X)) -> top(proper(X)) 24.08/8.29 top(ok(X)) -> top(active(X)) 24.08/8.29 24.08/8.29 S is empty. 24.08/8.29 Rewrite Strategy: FULL 24.25/8.33 EOF