28.50/8.83 WORST_CASE(Omega(n^1), O(n^1)) 28.50/8.84 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 28.50/8.84 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.50/8.84 28.50/8.84 28.50/8.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.50/8.84 28.50/8.84 (0) CpxTRS 28.50/8.84 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 6 ms] 28.50/8.84 (2) CpxTRS 28.50/8.84 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 28.50/8.84 (4) CpxTRS 28.50/8.84 (5) CpxTrsMatchBoundsTAProof [FINISHED, 50 ms] 28.50/8.84 (6) BOUNDS(1, n^1) 28.50/8.84 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 28.50/8.84 (8) TRS for Loop Detection 28.50/8.84 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 28.50/8.84 (10) BEST 28.50/8.84 (11) proven lower bound 28.50/8.84 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 28.50/8.84 (13) BOUNDS(n^1, INF) 28.50/8.84 (14) TRS for Loop Detection 28.50/8.84 28.50/8.84 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (0) 28.50/8.84 Obligation: 28.50/8.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.50/8.84 28.50/8.84 28.50/8.84 The TRS R consists of the following rules: 28.50/8.84 28.50/8.84 active(dbl(0)) -> mark(0) 28.50/8.84 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 28.50/8.84 active(dbls(nil)) -> mark(nil) 28.50/8.84 active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) 28.50/8.84 active(sel(0, cons(X, Y))) -> mark(X) 28.50/8.84 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 28.50/8.84 active(indx(nil, X)) -> mark(nil) 28.50/8.84 active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) 28.50/8.84 active(from(X)) -> mark(cons(X, from(s(X)))) 28.50/8.84 active(dbl(X)) -> dbl(active(X)) 28.50/8.84 active(dbls(X)) -> dbls(active(X)) 28.50/8.84 active(sel(X1, X2)) -> sel(active(X1), X2) 28.50/8.84 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.50/8.84 active(indx(X1, X2)) -> indx(active(X1), X2) 28.50/8.84 dbl(mark(X)) -> mark(dbl(X)) 28.50/8.84 dbls(mark(X)) -> mark(dbls(X)) 28.50/8.84 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.50/8.84 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.50/8.84 indx(mark(X1), X2) -> mark(indx(X1, X2)) 28.50/8.84 proper(dbl(X)) -> dbl(proper(X)) 28.50/8.84 proper(0) -> ok(0) 28.50/8.84 proper(s(X)) -> s(proper(X)) 28.50/8.84 proper(dbls(X)) -> dbls(proper(X)) 28.50/8.84 proper(nil) -> ok(nil) 28.50/8.84 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.50/8.84 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.50/8.84 proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) 28.50/8.84 proper(from(X)) -> from(proper(X)) 28.50/8.84 dbl(ok(X)) -> ok(dbl(X)) 28.50/8.84 s(ok(X)) -> ok(s(X)) 28.50/8.84 dbls(ok(X)) -> ok(dbls(X)) 28.50/8.84 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.50/8.84 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.50/8.84 indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) 28.50/8.84 from(ok(X)) -> ok(from(X)) 28.50/8.84 top(mark(X)) -> top(proper(X)) 28.50/8.84 top(ok(X)) -> top(active(X)) 28.50/8.84 28.50/8.84 S is empty. 28.50/8.84 Rewrite Strategy: FULL 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 28.50/8.84 The following defined symbols can occur below the 0th argument of top: proper, active 28.50/8.84 The following defined symbols can occur below the 0th argument of proper: proper, active 28.50/8.84 The following defined symbols can occur below the 0th argument of active: proper, active 28.50/8.84 28.50/8.84 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 28.50/8.84 active(dbl(0)) -> mark(0) 28.50/8.84 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 28.50/8.84 active(dbls(nil)) -> mark(nil) 28.50/8.84 active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) 28.50/8.84 active(sel(0, cons(X, Y))) -> mark(X) 28.50/8.84 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 28.50/8.84 active(indx(nil, X)) -> mark(nil) 28.50/8.84 active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) 28.50/8.84 active(from(X)) -> mark(cons(X, from(s(X)))) 28.50/8.84 active(dbl(X)) -> dbl(active(X)) 28.50/8.84 active(dbls(X)) -> dbls(active(X)) 28.50/8.84 active(sel(X1, X2)) -> sel(active(X1), X2) 28.50/8.84 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.50/8.84 active(indx(X1, X2)) -> indx(active(X1), X2) 28.50/8.84 proper(dbl(X)) -> dbl(proper(X)) 28.50/8.84 proper(s(X)) -> s(proper(X)) 28.50/8.84 proper(dbls(X)) -> dbls(proper(X)) 28.50/8.84 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.50/8.84 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.50/8.84 proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) 28.50/8.84 proper(from(X)) -> from(proper(X)) 28.50/8.84 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (2) 28.50/8.84 Obligation: 28.50/8.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 28.50/8.84 28.50/8.84 28.50/8.84 The TRS R consists of the following rules: 28.50/8.84 28.50/8.84 dbl(mark(X)) -> mark(dbl(X)) 28.50/8.84 dbls(mark(X)) -> mark(dbls(X)) 28.50/8.84 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.50/8.84 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.50/8.84 indx(mark(X1), X2) -> mark(indx(X1, X2)) 28.50/8.84 proper(0) -> ok(0) 28.50/8.84 proper(nil) -> ok(nil) 28.50/8.84 dbl(ok(X)) -> ok(dbl(X)) 28.50/8.84 s(ok(X)) -> ok(s(X)) 28.50/8.84 dbls(ok(X)) -> ok(dbls(X)) 28.50/8.84 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.50/8.84 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.50/8.84 indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) 28.50/8.84 from(ok(X)) -> ok(from(X)) 28.50/8.84 top(mark(X)) -> top(proper(X)) 28.50/8.84 top(ok(X)) -> top(active(X)) 28.50/8.84 28.50/8.84 S is empty. 28.50/8.84 Rewrite Strategy: FULL 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 28.50/8.84 transformed relative TRS to TRS 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (4) 28.50/8.84 Obligation: 28.50/8.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 28.50/8.84 28.50/8.84 28.50/8.84 The TRS R consists of the following rules: 28.50/8.84 28.50/8.84 dbl(mark(X)) -> mark(dbl(X)) 28.50/8.84 dbls(mark(X)) -> mark(dbls(X)) 28.50/8.84 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.50/8.84 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.50/8.84 indx(mark(X1), X2) -> mark(indx(X1, X2)) 28.50/8.84 proper(0) -> ok(0) 28.50/8.84 proper(nil) -> ok(nil) 28.50/8.84 dbl(ok(X)) -> ok(dbl(X)) 28.50/8.84 s(ok(X)) -> ok(s(X)) 28.50/8.84 dbls(ok(X)) -> ok(dbls(X)) 28.50/8.84 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.50/8.84 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.50/8.84 indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) 28.50/8.84 from(ok(X)) -> ok(from(X)) 28.50/8.84 top(mark(X)) -> top(proper(X)) 28.50/8.84 top(ok(X)) -> top(active(X)) 28.50/8.84 28.50/8.84 S is empty. 28.50/8.84 Rewrite Strategy: FULL 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (5) CpxTrsMatchBoundsTAProof (FINISHED) 28.50/8.84 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. 28.50/8.84 28.50/8.84 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 28.50/8.84 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9] 28.50/8.84 transitions: 28.50/8.84 mark0(0) -> 0 28.50/8.84 00() -> 0 28.50/8.84 ok0(0) -> 0 28.50/8.84 nil0() -> 0 28.50/8.84 active0(0) -> 0 28.50/8.84 dbl0(0) -> 1 28.50/8.84 dbls0(0) -> 2 28.50/8.84 sel0(0, 0) -> 3 28.50/8.84 indx0(0, 0) -> 4 28.50/8.84 proper0(0) -> 5 28.50/8.84 s0(0) -> 6 28.50/8.84 cons0(0, 0) -> 7 28.50/8.84 from0(0) -> 8 28.50/8.84 top0(0) -> 9 28.50/8.84 dbl1(0) -> 10 28.50/8.84 mark1(10) -> 1 28.50/8.84 dbls1(0) -> 11 28.50/8.84 mark1(11) -> 2 28.50/8.84 sel1(0, 0) -> 12 28.50/8.84 mark1(12) -> 3 28.50/8.84 indx1(0, 0) -> 13 28.50/8.84 mark1(13) -> 4 28.50/8.84 01() -> 14 28.50/8.84 ok1(14) -> 5 28.50/8.84 nil1() -> 15 28.50/8.84 ok1(15) -> 5 28.50/8.84 dbl1(0) -> 16 28.50/8.84 ok1(16) -> 1 28.50/8.84 s1(0) -> 17 28.50/8.84 ok1(17) -> 6 28.50/8.84 dbls1(0) -> 18 28.50/8.84 ok1(18) -> 2 28.50/8.84 cons1(0, 0) -> 19 28.50/8.84 ok1(19) -> 7 28.50/8.84 sel1(0, 0) -> 20 28.50/8.84 ok1(20) -> 3 28.50/8.84 indx1(0, 0) -> 21 28.50/8.84 ok1(21) -> 4 28.50/8.84 from1(0) -> 22 28.50/8.84 ok1(22) -> 8 28.50/8.84 proper1(0) -> 23 28.50/8.84 top1(23) -> 9 28.50/8.84 active1(0) -> 24 28.50/8.84 top1(24) -> 9 28.50/8.84 mark1(10) -> 10 28.50/8.84 mark1(10) -> 16 28.50/8.84 mark1(11) -> 11 28.50/8.84 mark1(11) -> 18 28.50/8.84 mark1(12) -> 12 28.50/8.84 mark1(12) -> 20 28.50/8.84 mark1(13) -> 13 28.50/8.84 mark1(13) -> 21 28.50/8.84 ok1(14) -> 23 28.50/8.84 ok1(15) -> 23 28.50/8.84 ok1(16) -> 10 28.50/8.84 ok1(16) -> 16 28.50/8.84 ok1(17) -> 17 28.50/8.84 ok1(18) -> 11 28.50/8.84 ok1(18) -> 18 28.50/8.84 ok1(19) -> 19 28.50/8.84 ok1(20) -> 12 28.50/8.84 ok1(20) -> 20 28.50/8.84 ok1(21) -> 13 28.50/8.84 ok1(21) -> 21 28.50/8.84 ok1(22) -> 22 28.50/8.84 active2(14) -> 25 28.50/8.84 top2(25) -> 9 28.50/8.84 active2(15) -> 25 28.50/8.84 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (6) 28.50/8.84 BOUNDS(1, n^1) 28.50/8.84 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 28.50/8.84 Transformed a relative TRS into a decreasing-loop problem. 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (8) 28.50/8.84 Obligation: 28.50/8.84 Analyzing the following TRS for decreasing loops: 28.50/8.84 28.50/8.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.50/8.84 28.50/8.84 28.50/8.84 The TRS R consists of the following rules: 28.50/8.84 28.50/8.84 active(dbl(0)) -> mark(0) 28.50/8.84 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 28.50/8.84 active(dbls(nil)) -> mark(nil) 28.50/8.84 active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) 28.50/8.84 active(sel(0, cons(X, Y))) -> mark(X) 28.50/8.84 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 28.50/8.84 active(indx(nil, X)) -> mark(nil) 28.50/8.84 active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) 28.50/8.84 active(from(X)) -> mark(cons(X, from(s(X)))) 28.50/8.84 active(dbl(X)) -> dbl(active(X)) 28.50/8.84 active(dbls(X)) -> dbls(active(X)) 28.50/8.84 active(sel(X1, X2)) -> sel(active(X1), X2) 28.50/8.84 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.50/8.84 active(indx(X1, X2)) -> indx(active(X1), X2) 28.50/8.84 dbl(mark(X)) -> mark(dbl(X)) 28.50/8.84 dbls(mark(X)) -> mark(dbls(X)) 28.50/8.84 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.50/8.84 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.50/8.84 indx(mark(X1), X2) -> mark(indx(X1, X2)) 28.50/8.84 proper(dbl(X)) -> dbl(proper(X)) 28.50/8.84 proper(0) -> ok(0) 28.50/8.84 proper(s(X)) -> s(proper(X)) 28.50/8.84 proper(dbls(X)) -> dbls(proper(X)) 28.50/8.84 proper(nil) -> ok(nil) 28.50/8.84 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.50/8.84 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.50/8.84 proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) 28.50/8.84 proper(from(X)) -> from(proper(X)) 28.50/8.84 dbl(ok(X)) -> ok(dbl(X)) 28.50/8.84 s(ok(X)) -> ok(s(X)) 28.50/8.84 dbls(ok(X)) -> ok(dbls(X)) 28.50/8.84 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.50/8.84 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.50/8.84 indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) 28.50/8.84 from(ok(X)) -> ok(from(X)) 28.50/8.84 top(mark(X)) -> top(proper(X)) 28.50/8.84 top(ok(X)) -> top(active(X)) 28.50/8.84 28.50/8.84 S is empty. 28.50/8.84 Rewrite Strategy: FULL 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (9) DecreasingLoopProof (LOWER BOUND(ID)) 28.50/8.84 The following loop(s) give(s) rise to the lower bound Omega(n^1): 28.50/8.84 28.50/8.84 The rewrite sequence 28.50/8.84 28.50/8.84 dbls(mark(X)) ->^+ mark(dbls(X)) 28.50/8.84 28.50/8.84 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 28.50/8.84 28.50/8.84 The pumping substitution is [X / mark(X)]. 28.50/8.84 28.50/8.84 The result substitution is [ ]. 28.50/8.84 28.50/8.84 28.50/8.84 28.50/8.84 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (10) 28.50/8.84 Complex Obligation (BEST) 28.50/8.84 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (11) 28.50/8.84 Obligation: 28.50/8.84 Proved the lower bound n^1 for the following obligation: 28.50/8.84 28.50/8.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.50/8.84 28.50/8.84 28.50/8.84 The TRS R consists of the following rules: 28.50/8.84 28.50/8.84 active(dbl(0)) -> mark(0) 28.50/8.84 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 28.50/8.84 active(dbls(nil)) -> mark(nil) 28.50/8.84 active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) 28.50/8.84 active(sel(0, cons(X, Y))) -> mark(X) 28.50/8.84 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 28.50/8.84 active(indx(nil, X)) -> mark(nil) 28.50/8.84 active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) 28.50/8.84 active(from(X)) -> mark(cons(X, from(s(X)))) 28.50/8.84 active(dbl(X)) -> dbl(active(X)) 28.50/8.84 active(dbls(X)) -> dbls(active(X)) 28.50/8.84 active(sel(X1, X2)) -> sel(active(X1), X2) 28.50/8.84 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.50/8.84 active(indx(X1, X2)) -> indx(active(X1), X2) 28.50/8.84 dbl(mark(X)) -> mark(dbl(X)) 28.50/8.84 dbls(mark(X)) -> mark(dbls(X)) 28.50/8.84 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.50/8.84 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.50/8.84 indx(mark(X1), X2) -> mark(indx(X1, X2)) 28.50/8.84 proper(dbl(X)) -> dbl(proper(X)) 28.50/8.84 proper(0) -> ok(0) 28.50/8.84 proper(s(X)) -> s(proper(X)) 28.50/8.84 proper(dbls(X)) -> dbls(proper(X)) 28.50/8.84 proper(nil) -> ok(nil) 28.50/8.84 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.50/8.84 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.50/8.84 proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) 28.50/8.84 proper(from(X)) -> from(proper(X)) 28.50/8.84 dbl(ok(X)) -> ok(dbl(X)) 28.50/8.84 s(ok(X)) -> ok(s(X)) 28.50/8.84 dbls(ok(X)) -> ok(dbls(X)) 28.50/8.84 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.50/8.84 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.50/8.84 indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) 28.50/8.84 from(ok(X)) -> ok(from(X)) 28.50/8.84 top(mark(X)) -> top(proper(X)) 28.50/8.84 top(ok(X)) -> top(active(X)) 28.50/8.84 28.50/8.84 S is empty. 28.50/8.84 Rewrite Strategy: FULL 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (12) LowerBoundPropagationProof (FINISHED) 28.50/8.84 Propagated lower bound. 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (13) 28.50/8.84 BOUNDS(n^1, INF) 28.50/8.84 28.50/8.84 ---------------------------------------- 28.50/8.84 28.50/8.84 (14) 28.50/8.84 Obligation: 28.50/8.84 Analyzing the following TRS for decreasing loops: 28.50/8.84 28.50/8.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.50/8.84 28.50/8.84 28.50/8.84 The TRS R consists of the following rules: 28.50/8.84 28.50/8.84 active(dbl(0)) -> mark(0) 28.50/8.84 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 28.50/8.84 active(dbls(nil)) -> mark(nil) 28.50/8.84 active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) 28.50/8.84 active(sel(0, cons(X, Y))) -> mark(X) 28.50/8.84 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 28.50/8.84 active(indx(nil, X)) -> mark(nil) 28.50/8.84 active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) 28.50/8.84 active(from(X)) -> mark(cons(X, from(s(X)))) 28.50/8.84 active(dbl(X)) -> dbl(active(X)) 28.50/8.84 active(dbls(X)) -> dbls(active(X)) 28.50/8.84 active(sel(X1, X2)) -> sel(active(X1), X2) 28.50/8.84 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.50/8.84 active(indx(X1, X2)) -> indx(active(X1), X2) 28.50/8.84 dbl(mark(X)) -> mark(dbl(X)) 28.50/8.84 dbls(mark(X)) -> mark(dbls(X)) 28.50/8.84 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.50/8.84 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.50/8.84 indx(mark(X1), X2) -> mark(indx(X1, X2)) 28.50/8.84 proper(dbl(X)) -> dbl(proper(X)) 28.50/8.84 proper(0) -> ok(0) 28.50/8.84 proper(s(X)) -> s(proper(X)) 28.50/8.84 proper(dbls(X)) -> dbls(proper(X)) 28.50/8.84 proper(nil) -> ok(nil) 28.50/8.84 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.50/8.84 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.50/8.84 proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) 28.50/8.84 proper(from(X)) -> from(proper(X)) 28.50/8.84 dbl(ok(X)) -> ok(dbl(X)) 28.50/8.84 s(ok(X)) -> ok(s(X)) 28.50/8.84 dbls(ok(X)) -> ok(dbls(X)) 28.50/8.84 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.50/8.84 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.50/8.84 indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) 28.50/8.84 from(ok(X)) -> ok(from(X)) 28.50/8.84 top(mark(X)) -> top(proper(X)) 28.50/8.84 top(ok(X)) -> top(active(X)) 28.50/8.84 28.50/8.84 S is empty. 28.50/8.84 Rewrite Strategy: FULL 28.56/8.89 EOF