24.93/8.51 WORST_CASE(Omega(n^1), O(n^1)) 24.93/8.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.93/8.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.93/8.52 24.93/8.52 24.93/8.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.93/8.52 24.93/8.52 (0) CpxTRS 24.93/8.52 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 24.93/8.52 (2) CpxTRS 24.93/8.52 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.93/8.52 (4) CpxTRS 24.93/8.52 (5) CpxTrsMatchBoundsTAProof [FINISHED, 61 ms] 24.93/8.52 (6) BOUNDS(1, n^1) 24.93/8.52 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 24.93/8.52 (8) TRS for Loop Detection 24.93/8.52 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 24.93/8.52 (10) BEST 24.93/8.52 (11) proven lower bound 24.93/8.52 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 24.93/8.52 (13) BOUNDS(n^1, INF) 24.93/8.52 (14) TRS for Loop Detection 24.93/8.52 24.93/8.52 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (0) 24.93/8.52 Obligation: 24.93/8.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.93/8.52 24.93/8.52 24.93/8.52 The TRS R consists of the following rules: 24.93/8.52 24.93/8.52 active(p(0)) -> mark(0) 24.93/8.52 active(p(s(X))) -> mark(X) 24.93/8.52 active(leq(0, Y)) -> mark(true) 24.93/8.52 active(leq(s(X), 0)) -> mark(false) 24.93/8.52 active(leq(s(X), s(Y))) -> mark(leq(X, Y)) 24.93/8.52 active(if(true, X, Y)) -> mark(X) 24.93/8.52 active(if(false, X, Y)) -> mark(Y) 24.93/8.52 active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) 24.93/8.52 active(p(X)) -> p(active(X)) 24.93/8.52 active(s(X)) -> s(active(X)) 24.93/8.52 active(leq(X1, X2)) -> leq(active(X1), X2) 24.93/8.52 active(leq(X1, X2)) -> leq(X1, active(X2)) 24.93/8.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.52 active(diff(X1, X2)) -> diff(active(X1), X2) 24.93/8.52 active(diff(X1, X2)) -> diff(X1, active(X2)) 24.93/8.52 p(mark(X)) -> mark(p(X)) 24.93/8.52 s(mark(X)) -> mark(s(X)) 24.93/8.52 leq(mark(X1), X2) -> mark(leq(X1, X2)) 24.93/8.52 leq(X1, mark(X2)) -> mark(leq(X1, X2)) 24.93/8.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.52 diff(mark(X1), X2) -> mark(diff(X1, X2)) 24.93/8.52 diff(X1, mark(X2)) -> mark(diff(X1, X2)) 24.93/8.52 proper(p(X)) -> p(proper(X)) 24.93/8.52 proper(0) -> ok(0) 24.93/8.52 proper(s(X)) -> s(proper(X)) 24.93/8.52 proper(leq(X1, X2)) -> leq(proper(X1), proper(X2)) 24.93/8.52 proper(true) -> ok(true) 24.93/8.52 proper(false) -> ok(false) 24.93/8.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.52 proper(diff(X1, X2)) -> diff(proper(X1), proper(X2)) 24.93/8.52 p(ok(X)) -> ok(p(X)) 24.93/8.52 s(ok(X)) -> ok(s(X)) 24.93/8.52 leq(ok(X1), ok(X2)) -> ok(leq(X1, X2)) 24.93/8.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.52 diff(ok(X1), ok(X2)) -> ok(diff(X1, X2)) 24.93/8.52 top(mark(X)) -> top(proper(X)) 24.93/8.52 top(ok(X)) -> top(active(X)) 24.93/8.52 24.93/8.52 S is empty. 24.93/8.52 Rewrite Strategy: FULL 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.93/8.52 The following defined symbols can occur below the 0th argument of top: proper, active 24.93/8.52 The following defined symbols can occur below the 0th argument of proper: proper, active 24.93/8.52 The following defined symbols can occur below the 0th argument of active: proper, active 24.93/8.52 24.93/8.52 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.93/8.52 active(p(0)) -> mark(0) 24.93/8.52 active(p(s(X))) -> mark(X) 24.93/8.52 active(leq(0, Y)) -> mark(true) 24.93/8.52 active(leq(s(X), 0)) -> mark(false) 24.93/8.52 active(leq(s(X), s(Y))) -> mark(leq(X, Y)) 24.93/8.52 active(if(true, X, Y)) -> mark(X) 24.93/8.52 active(if(false, X, Y)) -> mark(Y) 24.93/8.52 active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) 24.93/8.52 active(p(X)) -> p(active(X)) 24.93/8.52 active(s(X)) -> s(active(X)) 24.93/8.52 active(leq(X1, X2)) -> leq(active(X1), X2) 24.93/8.52 active(leq(X1, X2)) -> leq(X1, active(X2)) 24.93/8.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.52 active(diff(X1, X2)) -> diff(active(X1), X2) 24.93/8.52 active(diff(X1, X2)) -> diff(X1, active(X2)) 24.93/8.52 proper(p(X)) -> p(proper(X)) 24.93/8.52 proper(s(X)) -> s(proper(X)) 24.93/8.52 proper(leq(X1, X2)) -> leq(proper(X1), proper(X2)) 24.93/8.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.52 proper(diff(X1, X2)) -> diff(proper(X1), proper(X2)) 24.93/8.52 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (2) 24.93/8.52 Obligation: 24.93/8.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.93/8.52 24.93/8.52 24.93/8.52 The TRS R consists of the following rules: 24.93/8.52 24.93/8.52 p(mark(X)) -> mark(p(X)) 24.93/8.52 s(mark(X)) -> mark(s(X)) 24.93/8.52 leq(mark(X1), X2) -> mark(leq(X1, X2)) 24.93/8.52 leq(X1, mark(X2)) -> mark(leq(X1, X2)) 24.93/8.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.52 diff(mark(X1), X2) -> mark(diff(X1, X2)) 24.93/8.52 diff(X1, mark(X2)) -> mark(diff(X1, X2)) 24.93/8.52 proper(0) -> ok(0) 24.93/8.52 proper(true) -> ok(true) 24.93/8.52 proper(false) -> ok(false) 24.93/8.52 p(ok(X)) -> ok(p(X)) 24.93/8.52 s(ok(X)) -> ok(s(X)) 24.93/8.52 leq(ok(X1), ok(X2)) -> ok(leq(X1, X2)) 24.93/8.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.52 diff(ok(X1), ok(X2)) -> ok(diff(X1, X2)) 24.93/8.52 top(mark(X)) -> top(proper(X)) 24.93/8.52 top(ok(X)) -> top(active(X)) 24.93/8.52 24.93/8.52 S is empty. 24.93/8.52 Rewrite Strategy: FULL 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.93/8.52 transformed relative TRS to TRS 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (4) 24.93/8.52 Obligation: 24.93/8.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.93/8.52 24.93/8.52 24.93/8.52 The TRS R consists of the following rules: 24.93/8.52 24.93/8.52 p(mark(X)) -> mark(p(X)) 24.93/8.52 s(mark(X)) -> mark(s(X)) 24.93/8.52 leq(mark(X1), X2) -> mark(leq(X1, X2)) 24.93/8.52 leq(X1, mark(X2)) -> mark(leq(X1, X2)) 24.93/8.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.52 diff(mark(X1), X2) -> mark(diff(X1, X2)) 24.93/8.52 diff(X1, mark(X2)) -> mark(diff(X1, X2)) 24.93/8.52 proper(0) -> ok(0) 24.93/8.52 proper(true) -> ok(true) 24.93/8.52 proper(false) -> ok(false) 24.93/8.52 p(ok(X)) -> ok(p(X)) 24.93/8.52 s(ok(X)) -> ok(s(X)) 24.93/8.52 leq(ok(X1), ok(X2)) -> ok(leq(X1, X2)) 24.93/8.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.52 diff(ok(X1), ok(X2)) -> ok(diff(X1, X2)) 24.93/8.52 top(mark(X)) -> top(proper(X)) 24.93/8.52 top(ok(X)) -> top(active(X)) 24.93/8.52 24.93/8.52 S is empty. 24.93/8.52 Rewrite Strategy: FULL 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.93/8.52 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. 24.93/8.52 24.93/8.52 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.93/8.52 final states : [1, 2, 3, 4, 5, 6, 7] 24.93/8.52 transitions: 24.93/8.52 mark0(0) -> 0 24.93/8.52 00() -> 0 24.93/8.52 ok0(0) -> 0 24.93/8.52 true0() -> 0 24.93/8.52 false0() -> 0 24.93/8.52 active0(0) -> 0 24.93/8.52 p0(0) -> 1 24.93/8.52 s0(0) -> 2 24.93/8.52 leq0(0, 0) -> 3 24.93/8.52 if0(0, 0, 0) -> 4 24.93/8.52 diff0(0, 0) -> 5 24.93/8.52 proper0(0) -> 6 24.93/8.52 top0(0) -> 7 24.93/8.52 p1(0) -> 8 24.93/8.52 mark1(8) -> 1 24.93/8.52 s1(0) -> 9 24.93/8.52 mark1(9) -> 2 24.93/8.52 leq1(0, 0) -> 10 24.93/8.52 mark1(10) -> 3 24.93/8.52 if1(0, 0, 0) -> 11 24.93/8.52 mark1(11) -> 4 24.93/8.52 diff1(0, 0) -> 12 24.93/8.52 mark1(12) -> 5 24.93/8.52 01() -> 13 24.93/8.52 ok1(13) -> 6 24.93/8.52 true1() -> 14 24.93/8.52 ok1(14) -> 6 24.93/8.52 false1() -> 15 24.93/8.52 ok1(15) -> 6 24.93/8.52 p1(0) -> 16 24.93/8.52 ok1(16) -> 1 24.93/8.52 s1(0) -> 17 24.93/8.52 ok1(17) -> 2 24.93/8.52 leq1(0, 0) -> 18 24.93/8.52 ok1(18) -> 3 24.93/8.52 if1(0, 0, 0) -> 19 24.93/8.52 ok1(19) -> 4 24.93/8.52 diff1(0, 0) -> 20 24.93/8.52 ok1(20) -> 5 24.93/8.52 proper1(0) -> 21 24.93/8.52 top1(21) -> 7 24.93/8.52 active1(0) -> 22 24.93/8.52 top1(22) -> 7 24.93/8.52 mark1(8) -> 8 24.93/8.52 mark1(8) -> 16 24.93/8.52 mark1(9) -> 9 24.93/8.52 mark1(9) -> 17 24.93/8.52 mark1(10) -> 10 24.93/8.52 mark1(10) -> 18 24.93/8.52 mark1(11) -> 11 24.93/8.52 mark1(11) -> 19 24.93/8.52 mark1(12) -> 12 24.93/8.52 mark1(12) -> 20 24.93/8.52 ok1(13) -> 21 24.93/8.52 ok1(14) -> 21 24.93/8.52 ok1(15) -> 21 24.93/8.52 ok1(16) -> 8 24.93/8.52 ok1(16) -> 16 24.93/8.52 ok1(17) -> 9 24.93/8.52 ok1(17) -> 17 24.93/8.52 ok1(18) -> 10 24.93/8.52 ok1(18) -> 18 24.93/8.52 ok1(19) -> 11 24.93/8.52 ok1(19) -> 19 24.93/8.52 ok1(20) -> 12 24.93/8.52 ok1(20) -> 20 24.93/8.52 active2(13) -> 23 24.93/8.52 top2(23) -> 7 24.93/8.52 active2(14) -> 23 24.93/8.52 active2(15) -> 23 24.93/8.52 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (6) 24.93/8.52 BOUNDS(1, n^1) 24.93/8.52 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 24.93/8.52 Transformed a relative TRS into a decreasing-loop problem. 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (8) 24.93/8.52 Obligation: 24.93/8.52 Analyzing the following TRS for decreasing loops: 24.93/8.52 24.93/8.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.93/8.52 24.93/8.52 24.93/8.52 The TRS R consists of the following rules: 24.93/8.52 24.93/8.52 active(p(0)) -> mark(0) 24.93/8.52 active(p(s(X))) -> mark(X) 24.93/8.52 active(leq(0, Y)) -> mark(true) 24.93/8.52 active(leq(s(X), 0)) -> mark(false) 24.93/8.52 active(leq(s(X), s(Y))) -> mark(leq(X, Y)) 24.93/8.52 active(if(true, X, Y)) -> mark(X) 24.93/8.52 active(if(false, X, Y)) -> mark(Y) 24.93/8.52 active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) 24.93/8.52 active(p(X)) -> p(active(X)) 24.93/8.52 active(s(X)) -> s(active(X)) 24.93/8.52 active(leq(X1, X2)) -> leq(active(X1), X2) 24.93/8.52 active(leq(X1, X2)) -> leq(X1, active(X2)) 24.93/8.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.52 active(diff(X1, X2)) -> diff(active(X1), X2) 24.93/8.52 active(diff(X1, X2)) -> diff(X1, active(X2)) 24.93/8.52 p(mark(X)) -> mark(p(X)) 24.93/8.52 s(mark(X)) -> mark(s(X)) 24.93/8.52 leq(mark(X1), X2) -> mark(leq(X1, X2)) 24.93/8.52 leq(X1, mark(X2)) -> mark(leq(X1, X2)) 24.93/8.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.52 diff(mark(X1), X2) -> mark(diff(X1, X2)) 24.93/8.52 diff(X1, mark(X2)) -> mark(diff(X1, X2)) 24.93/8.52 proper(p(X)) -> p(proper(X)) 24.93/8.52 proper(0) -> ok(0) 24.93/8.52 proper(s(X)) -> s(proper(X)) 24.93/8.52 proper(leq(X1, X2)) -> leq(proper(X1), proper(X2)) 24.93/8.52 proper(true) -> ok(true) 24.93/8.52 proper(false) -> ok(false) 24.93/8.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.52 proper(diff(X1, X2)) -> diff(proper(X1), proper(X2)) 24.93/8.52 p(ok(X)) -> ok(p(X)) 24.93/8.52 s(ok(X)) -> ok(s(X)) 24.93/8.52 leq(ok(X1), ok(X2)) -> ok(leq(X1, X2)) 24.93/8.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.52 diff(ok(X1), ok(X2)) -> ok(diff(X1, X2)) 24.93/8.52 top(mark(X)) -> top(proper(X)) 24.93/8.52 top(ok(X)) -> top(active(X)) 24.93/8.52 24.93/8.52 S is empty. 24.93/8.52 Rewrite Strategy: FULL 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (9) DecreasingLoopProof (LOWER BOUND(ID)) 24.93/8.52 The following loop(s) give(s) rise to the lower bound Omega(n^1): 24.93/8.52 24.93/8.52 The rewrite sequence 24.93/8.52 24.93/8.52 p(mark(X)) ->^+ mark(p(X)) 24.93/8.52 24.93/8.52 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 24.93/8.52 24.93/8.52 The pumping substitution is [X / mark(X)]. 24.93/8.52 24.93/8.52 The result substitution is [ ]. 24.93/8.52 24.93/8.52 24.93/8.52 24.93/8.52 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (10) 24.93/8.52 Complex Obligation (BEST) 24.93/8.52 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (11) 24.93/8.52 Obligation: 24.93/8.52 Proved the lower bound n^1 for the following obligation: 24.93/8.52 24.93/8.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.93/8.52 24.93/8.52 24.93/8.52 The TRS R consists of the following rules: 24.93/8.52 24.93/8.52 active(p(0)) -> mark(0) 24.93/8.52 active(p(s(X))) -> mark(X) 24.93/8.52 active(leq(0, Y)) -> mark(true) 24.93/8.52 active(leq(s(X), 0)) -> mark(false) 24.93/8.52 active(leq(s(X), s(Y))) -> mark(leq(X, Y)) 24.93/8.52 active(if(true, X, Y)) -> mark(X) 24.93/8.52 active(if(false, X, Y)) -> mark(Y) 24.93/8.52 active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) 24.93/8.52 active(p(X)) -> p(active(X)) 24.93/8.52 active(s(X)) -> s(active(X)) 24.93/8.52 active(leq(X1, X2)) -> leq(active(X1), X2) 24.93/8.52 active(leq(X1, X2)) -> leq(X1, active(X2)) 24.93/8.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.52 active(diff(X1, X2)) -> diff(active(X1), X2) 24.93/8.52 active(diff(X1, X2)) -> diff(X1, active(X2)) 24.93/8.52 p(mark(X)) -> mark(p(X)) 24.93/8.52 s(mark(X)) -> mark(s(X)) 24.93/8.52 leq(mark(X1), X2) -> mark(leq(X1, X2)) 24.93/8.52 leq(X1, mark(X2)) -> mark(leq(X1, X2)) 24.93/8.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.52 diff(mark(X1), X2) -> mark(diff(X1, X2)) 24.93/8.52 diff(X1, mark(X2)) -> mark(diff(X1, X2)) 24.93/8.52 proper(p(X)) -> p(proper(X)) 24.93/8.52 proper(0) -> ok(0) 24.93/8.52 proper(s(X)) -> s(proper(X)) 24.93/8.52 proper(leq(X1, X2)) -> leq(proper(X1), proper(X2)) 24.93/8.52 proper(true) -> ok(true) 24.93/8.52 proper(false) -> ok(false) 24.93/8.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.52 proper(diff(X1, X2)) -> diff(proper(X1), proper(X2)) 24.93/8.52 p(ok(X)) -> ok(p(X)) 24.93/8.52 s(ok(X)) -> ok(s(X)) 24.93/8.52 leq(ok(X1), ok(X2)) -> ok(leq(X1, X2)) 24.93/8.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.52 diff(ok(X1), ok(X2)) -> ok(diff(X1, X2)) 24.93/8.52 top(mark(X)) -> top(proper(X)) 24.93/8.52 top(ok(X)) -> top(active(X)) 24.93/8.52 24.93/8.52 S is empty. 24.93/8.52 Rewrite Strategy: FULL 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (12) LowerBoundPropagationProof (FINISHED) 24.93/8.52 Propagated lower bound. 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (13) 24.93/8.52 BOUNDS(n^1, INF) 24.93/8.52 24.93/8.52 ---------------------------------------- 24.93/8.52 24.93/8.52 (14) 24.93/8.52 Obligation: 24.93/8.52 Analyzing the following TRS for decreasing loops: 24.93/8.52 24.93/8.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.93/8.52 24.93/8.52 24.93/8.52 The TRS R consists of the following rules: 24.93/8.52 24.93/8.52 active(p(0)) -> mark(0) 24.93/8.52 active(p(s(X))) -> mark(X) 24.93/8.52 active(leq(0, Y)) -> mark(true) 24.93/8.52 active(leq(s(X), 0)) -> mark(false) 24.93/8.52 active(leq(s(X), s(Y))) -> mark(leq(X, Y)) 24.93/8.52 active(if(true, X, Y)) -> mark(X) 24.93/8.52 active(if(false, X, Y)) -> mark(Y) 24.93/8.52 active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) 24.93/8.52 active(p(X)) -> p(active(X)) 24.93/8.52 active(s(X)) -> s(active(X)) 24.93/8.52 active(leq(X1, X2)) -> leq(active(X1), X2) 24.93/8.52 active(leq(X1, X2)) -> leq(X1, active(X2)) 24.93/8.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.52 active(diff(X1, X2)) -> diff(active(X1), X2) 24.93/8.52 active(diff(X1, X2)) -> diff(X1, active(X2)) 24.93/8.52 p(mark(X)) -> mark(p(X)) 24.93/8.52 s(mark(X)) -> mark(s(X)) 24.93/8.52 leq(mark(X1), X2) -> mark(leq(X1, X2)) 24.93/8.52 leq(X1, mark(X2)) -> mark(leq(X1, X2)) 24.93/8.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.52 diff(mark(X1), X2) -> mark(diff(X1, X2)) 24.93/8.52 diff(X1, mark(X2)) -> mark(diff(X1, X2)) 24.93/8.52 proper(p(X)) -> p(proper(X)) 24.93/8.52 proper(0) -> ok(0) 24.93/8.52 proper(s(X)) -> s(proper(X)) 24.93/8.52 proper(leq(X1, X2)) -> leq(proper(X1), proper(X2)) 24.93/8.52 proper(true) -> ok(true) 24.93/8.52 proper(false) -> ok(false) 24.93/8.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.52 proper(diff(X1, X2)) -> diff(proper(X1), proper(X2)) 24.93/8.52 p(ok(X)) -> ok(p(X)) 24.93/8.52 s(ok(X)) -> ok(s(X)) 24.93/8.52 leq(ok(X1), ok(X2)) -> ok(leq(X1, X2)) 24.93/8.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.52 diff(ok(X1), ok(X2)) -> ok(diff(X1, X2)) 24.93/8.52 top(mark(X)) -> top(proper(X)) 24.93/8.52 top(ok(X)) -> top(active(X)) 24.93/8.52 24.93/8.52 S is empty. 24.93/8.52 Rewrite Strategy: FULL 25.04/8.57 EOF