32.24/9.49 WORST_CASE(Omega(n^1), O(n^1)) 32.46/9.50 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 32.46/9.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 32.46/9.50 32.46/9.50 32.46/9.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 32.46/9.50 32.46/9.50 (0) CpxTRS 32.46/9.50 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 13 ms] 32.46/9.50 (2) CpxTRS 32.46/9.50 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 32.46/9.50 (4) CpxTRS 32.46/9.50 (5) CpxTrsMatchBoundsTAProof [FINISHED, 85 ms] 32.46/9.50 (6) BOUNDS(1, n^1) 32.46/9.50 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 32.46/9.50 (8) TRS for Loop Detection 32.46/9.50 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 32.46/9.50 (10) BEST 32.46/9.50 (11) proven lower bound 32.46/9.50 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 32.46/9.50 (13) BOUNDS(n^1, INF) 32.46/9.50 (14) TRS for Loop Detection 32.46/9.50 32.46/9.50 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (0) 32.46/9.50 Obligation: 32.46/9.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 32.46/9.50 32.46/9.50 32.46/9.50 The TRS R consists of the following rules: 32.46/9.50 32.46/9.50 active(U11(tt, N)) -> mark(N) 32.46/9.50 active(U21(tt, M, N)) -> mark(s(plus(N, M))) 32.46/9.50 active(and(tt, X)) -> mark(X) 32.46/9.50 active(isNat(0)) -> mark(tt) 32.46/9.50 active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) 32.46/9.50 active(isNat(s(V1))) -> mark(isNat(V1)) 32.46/9.50 active(plus(N, 0)) -> mark(U11(isNat(N), N)) 32.46/9.50 active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) 32.46/9.50 active(U11(X1, X2)) -> U11(active(X1), X2) 32.46/9.50 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 32.46/9.50 active(s(X)) -> s(active(X)) 32.46/9.50 active(plus(X1, X2)) -> plus(active(X1), X2) 32.46/9.50 active(plus(X1, X2)) -> plus(X1, active(X2)) 32.46/9.50 active(and(X1, X2)) -> and(active(X1), X2) 32.46/9.50 U11(mark(X1), X2) -> mark(U11(X1, X2)) 32.46/9.50 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 32.46/9.50 s(mark(X)) -> mark(s(X)) 32.46/9.50 plus(mark(X1), X2) -> mark(plus(X1, X2)) 32.46/9.50 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 32.46/9.50 and(mark(X1), X2) -> mark(and(X1, X2)) 32.46/9.50 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 32.46/9.50 proper(tt) -> ok(tt) 32.46/9.50 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 32.46/9.50 proper(s(X)) -> s(proper(X)) 32.46/9.50 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 32.46/9.50 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 32.46/9.50 proper(isNat(X)) -> isNat(proper(X)) 32.46/9.50 proper(0) -> ok(0) 32.46/9.50 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 32.46/9.50 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 32.46/9.50 s(ok(X)) -> ok(s(X)) 32.46/9.50 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 32.46/9.50 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 32.46/9.50 isNat(ok(X)) -> ok(isNat(X)) 32.46/9.50 top(mark(X)) -> top(proper(X)) 32.46/9.50 top(ok(X)) -> top(active(X)) 32.46/9.50 32.46/9.50 S is empty. 32.46/9.50 Rewrite Strategy: FULL 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 32.46/9.50 The following defined symbols can occur below the 0th argument of top: proper, active 32.46/9.50 The following defined symbols can occur below the 0th argument of proper: proper, active 32.46/9.50 The following defined symbols can occur below the 0th argument of active: proper, active 32.46/9.50 32.46/9.50 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 32.46/9.50 active(U11(tt, N)) -> mark(N) 32.46/9.50 active(U21(tt, M, N)) -> mark(s(plus(N, M))) 32.46/9.50 active(and(tt, X)) -> mark(X) 32.46/9.50 active(isNat(0)) -> mark(tt) 32.46/9.50 active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) 32.46/9.50 active(isNat(s(V1))) -> mark(isNat(V1)) 32.46/9.50 active(plus(N, 0)) -> mark(U11(isNat(N), N)) 32.46/9.50 active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) 32.46/9.50 active(U11(X1, X2)) -> U11(active(X1), X2) 32.46/9.50 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 32.46/9.50 active(s(X)) -> s(active(X)) 32.46/9.50 active(plus(X1, X2)) -> plus(active(X1), X2) 32.46/9.50 active(plus(X1, X2)) -> plus(X1, active(X2)) 32.46/9.50 active(and(X1, X2)) -> and(active(X1), X2) 32.46/9.50 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 32.46/9.50 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 32.46/9.50 proper(s(X)) -> s(proper(X)) 32.46/9.50 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 32.46/9.50 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 32.46/9.50 proper(isNat(X)) -> isNat(proper(X)) 32.46/9.50 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (2) 32.46/9.50 Obligation: 32.46/9.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 32.46/9.50 32.46/9.50 32.46/9.50 The TRS R consists of the following rules: 32.46/9.50 32.46/9.50 U11(mark(X1), X2) -> mark(U11(X1, X2)) 32.46/9.50 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 32.46/9.50 s(mark(X)) -> mark(s(X)) 32.46/9.50 plus(mark(X1), X2) -> mark(plus(X1, X2)) 32.46/9.50 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 32.46/9.50 and(mark(X1), X2) -> mark(and(X1, X2)) 32.46/9.50 proper(tt) -> ok(tt) 32.46/9.50 proper(0) -> ok(0) 32.46/9.50 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 32.46/9.50 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 32.46/9.50 s(ok(X)) -> ok(s(X)) 32.46/9.50 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 32.46/9.50 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 32.46/9.50 isNat(ok(X)) -> ok(isNat(X)) 32.46/9.50 top(mark(X)) -> top(proper(X)) 32.46/9.50 top(ok(X)) -> top(active(X)) 32.46/9.50 32.46/9.50 S is empty. 32.46/9.50 Rewrite Strategy: FULL 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 32.46/9.50 transformed relative TRS to TRS 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (4) 32.46/9.50 Obligation: 32.46/9.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 32.46/9.50 32.46/9.50 32.46/9.50 The TRS R consists of the following rules: 32.46/9.50 32.46/9.50 U11(mark(X1), X2) -> mark(U11(X1, X2)) 32.46/9.50 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 32.46/9.50 s(mark(X)) -> mark(s(X)) 32.46/9.50 plus(mark(X1), X2) -> mark(plus(X1, X2)) 32.46/9.50 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 32.46/9.50 and(mark(X1), X2) -> mark(and(X1, X2)) 32.46/9.50 proper(tt) -> ok(tt) 32.46/9.50 proper(0) -> ok(0) 32.46/9.50 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 32.46/9.50 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 32.46/9.50 s(ok(X)) -> ok(s(X)) 32.46/9.50 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 32.46/9.50 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 32.46/9.50 isNat(ok(X)) -> ok(isNat(X)) 32.46/9.50 top(mark(X)) -> top(proper(X)) 32.46/9.50 top(ok(X)) -> top(active(X)) 32.46/9.50 32.46/9.50 S is empty. 32.46/9.50 Rewrite Strategy: FULL 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (5) CpxTrsMatchBoundsTAProof (FINISHED) 32.46/9.50 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. 32.46/9.50 32.46/9.50 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 32.46/9.50 final states : [1, 2, 3, 4, 5, 6, 7, 8] 32.46/9.50 transitions: 32.46/9.50 mark0(0) -> 0 32.46/9.50 tt0() -> 0 32.46/9.50 ok0(0) -> 0 32.46/9.50 00() -> 0 32.46/9.50 active0(0) -> 0 32.46/9.50 U110(0, 0) -> 1 32.46/9.50 U210(0, 0, 0) -> 2 32.46/9.50 s0(0) -> 3 32.46/9.50 plus0(0, 0) -> 4 32.46/9.50 and0(0, 0) -> 5 32.46/9.50 proper0(0) -> 6 32.46/9.50 isNat0(0) -> 7 32.46/9.50 top0(0) -> 8 32.46/9.50 U111(0, 0) -> 9 32.46/9.50 mark1(9) -> 1 32.46/9.50 U211(0, 0, 0) -> 10 32.46/9.50 mark1(10) -> 2 32.46/9.50 s1(0) -> 11 32.46/9.50 mark1(11) -> 3 32.46/9.50 plus1(0, 0) -> 12 32.46/9.50 mark1(12) -> 4 32.46/9.50 and1(0, 0) -> 13 32.46/9.50 mark1(13) -> 5 32.46/9.50 tt1() -> 14 32.46/9.50 ok1(14) -> 6 32.46/9.50 01() -> 15 32.46/9.50 ok1(15) -> 6 32.46/9.50 U111(0, 0) -> 16 32.46/9.50 ok1(16) -> 1 32.46/9.50 U211(0, 0, 0) -> 17 32.46/9.50 ok1(17) -> 2 32.46/9.50 s1(0) -> 18 32.46/9.50 ok1(18) -> 3 32.46/9.50 plus1(0, 0) -> 19 32.46/9.50 ok1(19) -> 4 32.46/9.50 and1(0, 0) -> 20 32.46/9.50 ok1(20) -> 5 32.46/9.50 isNat1(0) -> 21 32.46/9.50 ok1(21) -> 7 32.46/9.50 proper1(0) -> 22 32.46/9.50 top1(22) -> 8 32.46/9.50 active1(0) -> 23 32.46/9.50 top1(23) -> 8 32.46/9.50 mark1(9) -> 9 32.46/9.50 mark1(9) -> 16 32.46/9.50 mark1(10) -> 10 32.46/9.50 mark1(10) -> 17 32.46/9.50 mark1(11) -> 11 32.46/9.50 mark1(11) -> 18 32.46/9.50 mark1(12) -> 12 32.46/9.50 mark1(12) -> 19 32.46/9.50 mark1(13) -> 13 32.46/9.50 mark1(13) -> 20 32.46/9.50 ok1(14) -> 22 32.46/9.50 ok1(15) -> 22 32.46/9.50 ok1(16) -> 9 32.46/9.50 ok1(16) -> 16 32.46/9.50 ok1(17) -> 10 32.46/9.50 ok1(17) -> 17 32.46/9.50 ok1(18) -> 11 32.46/9.50 ok1(18) -> 18 32.46/9.50 ok1(19) -> 12 32.46/9.50 ok1(19) -> 19 32.46/9.50 ok1(20) -> 13 32.46/9.50 ok1(20) -> 20 32.46/9.50 ok1(21) -> 21 32.46/9.50 active2(14) -> 24 32.46/9.50 top2(24) -> 8 32.46/9.50 active2(15) -> 24 32.46/9.50 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (6) 32.46/9.50 BOUNDS(1, n^1) 32.46/9.50 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 32.46/9.50 Transformed a relative TRS into a decreasing-loop problem. 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (8) 32.46/9.50 Obligation: 32.46/9.50 Analyzing the following TRS for decreasing loops: 32.46/9.50 32.46/9.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 32.46/9.50 32.46/9.50 32.46/9.50 The TRS R consists of the following rules: 32.46/9.50 32.46/9.50 active(U11(tt, N)) -> mark(N) 32.46/9.50 active(U21(tt, M, N)) -> mark(s(plus(N, M))) 32.46/9.50 active(and(tt, X)) -> mark(X) 32.46/9.50 active(isNat(0)) -> mark(tt) 32.46/9.50 active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) 32.46/9.50 active(isNat(s(V1))) -> mark(isNat(V1)) 32.46/9.50 active(plus(N, 0)) -> mark(U11(isNat(N), N)) 32.46/9.50 active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) 32.46/9.50 active(U11(X1, X2)) -> U11(active(X1), X2) 32.46/9.50 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 32.46/9.50 active(s(X)) -> s(active(X)) 32.46/9.50 active(plus(X1, X2)) -> plus(active(X1), X2) 32.46/9.50 active(plus(X1, X2)) -> plus(X1, active(X2)) 32.46/9.50 active(and(X1, X2)) -> and(active(X1), X2) 32.46/9.50 U11(mark(X1), X2) -> mark(U11(X1, X2)) 32.46/9.50 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 32.46/9.50 s(mark(X)) -> mark(s(X)) 32.46/9.50 plus(mark(X1), X2) -> mark(plus(X1, X2)) 32.46/9.50 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 32.46/9.50 and(mark(X1), X2) -> mark(and(X1, X2)) 32.46/9.50 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 32.46/9.50 proper(tt) -> ok(tt) 32.46/9.50 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 32.46/9.50 proper(s(X)) -> s(proper(X)) 32.46/9.50 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 32.46/9.50 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 32.46/9.50 proper(isNat(X)) -> isNat(proper(X)) 32.46/9.50 proper(0) -> ok(0) 32.46/9.50 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 32.46/9.50 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 32.46/9.50 s(ok(X)) -> ok(s(X)) 32.46/9.50 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 32.46/9.50 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 32.46/9.50 isNat(ok(X)) -> ok(isNat(X)) 32.46/9.50 top(mark(X)) -> top(proper(X)) 32.46/9.50 top(ok(X)) -> top(active(X)) 32.46/9.50 32.46/9.50 S is empty. 32.46/9.50 Rewrite Strategy: FULL 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (9) DecreasingLoopProof (LOWER BOUND(ID)) 32.46/9.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 32.46/9.50 32.46/9.50 The rewrite sequence 32.46/9.50 32.46/9.50 plus(X1, mark(X2)) ->^+ mark(plus(X1, X2)) 32.46/9.50 32.46/9.50 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 32.46/9.50 32.46/9.50 The pumping substitution is [X2 / mark(X2)]. 32.46/9.50 32.46/9.50 The result substitution is [ ]. 32.46/9.50 32.46/9.50 32.46/9.50 32.46/9.50 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (10) 32.46/9.50 Complex Obligation (BEST) 32.46/9.50 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (11) 32.46/9.50 Obligation: 32.46/9.50 Proved the lower bound n^1 for the following obligation: 32.46/9.50 32.46/9.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 32.46/9.50 32.46/9.50 32.46/9.50 The TRS R consists of the following rules: 32.46/9.50 32.46/9.50 active(U11(tt, N)) -> mark(N) 32.46/9.50 active(U21(tt, M, N)) -> mark(s(plus(N, M))) 32.46/9.50 active(and(tt, X)) -> mark(X) 32.46/9.50 active(isNat(0)) -> mark(tt) 32.46/9.50 active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) 32.46/9.50 active(isNat(s(V1))) -> mark(isNat(V1)) 32.46/9.50 active(plus(N, 0)) -> mark(U11(isNat(N), N)) 32.46/9.50 active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) 32.46/9.50 active(U11(X1, X2)) -> U11(active(X1), X2) 32.46/9.50 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 32.46/9.50 active(s(X)) -> s(active(X)) 32.46/9.50 active(plus(X1, X2)) -> plus(active(X1), X2) 32.46/9.50 active(plus(X1, X2)) -> plus(X1, active(X2)) 32.46/9.50 active(and(X1, X2)) -> and(active(X1), X2) 32.46/9.50 U11(mark(X1), X2) -> mark(U11(X1, X2)) 32.46/9.50 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 32.46/9.50 s(mark(X)) -> mark(s(X)) 32.46/9.50 plus(mark(X1), X2) -> mark(plus(X1, X2)) 32.46/9.50 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 32.46/9.50 and(mark(X1), X2) -> mark(and(X1, X2)) 32.46/9.50 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 32.46/9.50 proper(tt) -> ok(tt) 32.46/9.50 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 32.46/9.50 proper(s(X)) -> s(proper(X)) 32.46/9.50 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 32.46/9.50 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 32.46/9.50 proper(isNat(X)) -> isNat(proper(X)) 32.46/9.50 proper(0) -> ok(0) 32.46/9.50 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 32.46/9.50 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 32.46/9.50 s(ok(X)) -> ok(s(X)) 32.46/9.50 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 32.46/9.50 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 32.46/9.50 isNat(ok(X)) -> ok(isNat(X)) 32.46/9.50 top(mark(X)) -> top(proper(X)) 32.46/9.50 top(ok(X)) -> top(active(X)) 32.46/9.50 32.46/9.50 S is empty. 32.46/9.50 Rewrite Strategy: FULL 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (12) LowerBoundPropagationProof (FINISHED) 32.46/9.50 Propagated lower bound. 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (13) 32.46/9.50 BOUNDS(n^1, INF) 32.46/9.50 32.46/9.50 ---------------------------------------- 32.46/9.50 32.46/9.50 (14) 32.46/9.50 Obligation: 32.46/9.50 Analyzing the following TRS for decreasing loops: 32.46/9.50 32.46/9.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 32.46/9.50 32.46/9.50 32.46/9.50 The TRS R consists of the following rules: 32.46/9.50 32.46/9.50 active(U11(tt, N)) -> mark(N) 32.46/9.50 active(U21(tt, M, N)) -> mark(s(plus(N, M))) 32.46/9.50 active(and(tt, X)) -> mark(X) 32.46/9.50 active(isNat(0)) -> mark(tt) 32.46/9.50 active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) 32.46/9.50 active(isNat(s(V1))) -> mark(isNat(V1)) 32.46/9.50 active(plus(N, 0)) -> mark(U11(isNat(N), N)) 32.46/9.50 active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) 32.46/9.50 active(U11(X1, X2)) -> U11(active(X1), X2) 32.46/9.50 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 32.46/9.50 active(s(X)) -> s(active(X)) 32.46/9.50 active(plus(X1, X2)) -> plus(active(X1), X2) 32.46/9.50 active(plus(X1, X2)) -> plus(X1, active(X2)) 32.46/9.50 active(and(X1, X2)) -> and(active(X1), X2) 32.46/9.50 U11(mark(X1), X2) -> mark(U11(X1, X2)) 32.46/9.50 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 32.46/9.50 s(mark(X)) -> mark(s(X)) 32.46/9.50 plus(mark(X1), X2) -> mark(plus(X1, X2)) 32.46/9.50 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 32.46/9.50 and(mark(X1), X2) -> mark(and(X1, X2)) 32.46/9.50 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 32.46/9.50 proper(tt) -> ok(tt) 32.46/9.50 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 32.46/9.50 proper(s(X)) -> s(proper(X)) 32.46/9.50 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 32.46/9.50 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 32.46/9.50 proper(isNat(X)) -> isNat(proper(X)) 32.46/9.50 proper(0) -> ok(0) 32.46/9.50 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 32.46/9.50 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 32.46/9.50 s(ok(X)) -> ok(s(X)) 32.46/9.50 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 32.46/9.50 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 32.46/9.50 isNat(ok(X)) -> ok(isNat(X)) 32.46/9.50 top(mark(X)) -> top(proper(X)) 32.46/9.50 top(ok(X)) -> top(active(X)) 32.46/9.50 32.46/9.50 S is empty. 32.46/9.50 Rewrite Strategy: FULL 32.51/12.80 EOF