26.78/8.47 WORST_CASE(Omega(n^1), O(n^1)) 26.78/8.48 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 26.78/8.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.78/8.48 26.78/8.48 26.78/8.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 26.78/8.48 26.78/8.48 (0) CpxTRS 26.78/8.48 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 7 ms] 26.78/8.48 (2) CpxTRS 26.78/8.48 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 26.78/8.48 (4) CpxTRS 26.78/8.48 (5) CpxTrsMatchBoundsTAProof [FINISHED, 123 ms] 26.78/8.48 (6) BOUNDS(1, n^1) 26.78/8.48 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 26.78/8.48 (8) TRS for Loop Detection 26.78/8.48 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 26.78/8.48 (10) BEST 26.78/8.48 (11) proven lower bound 26.78/8.48 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 26.78/8.48 (13) BOUNDS(n^1, INF) 26.78/8.48 (14) TRS for Loop Detection 26.78/8.48 26.78/8.48 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (0) 26.78/8.48 Obligation: 26.78/8.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 26.78/8.48 26.78/8.48 26.78/8.48 The TRS R consists of the following rules: 26.78/8.48 26.78/8.48 active(and(true, X)) -> mark(X) 26.78/8.48 active(and(false, Y)) -> mark(false) 26.78/8.48 active(if(true, X, Y)) -> mark(X) 26.78/8.48 active(if(false, X, Y)) -> mark(Y) 26.78/8.48 active(add(0, X)) -> mark(X) 26.78/8.48 active(add(s(X), Y)) -> mark(s(add(X, Y))) 26.78/8.48 active(first(0, X)) -> mark(nil) 26.78/8.48 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 26.78/8.48 active(from(X)) -> mark(cons(X, from(s(X)))) 26.78/8.48 active(and(X1, X2)) -> and(active(X1), X2) 26.78/8.48 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 26.78/8.48 active(add(X1, X2)) -> add(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(X1, active(X2)) 26.78/8.48 and(mark(X1), X2) -> mark(and(X1, X2)) 26.78/8.48 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 26.78/8.48 add(mark(X1), X2) -> mark(add(X1, X2)) 26.78/8.48 first(mark(X1), X2) -> mark(first(X1, X2)) 26.78/8.48 first(X1, mark(X2)) -> mark(first(X1, X2)) 26.78/8.48 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 26.78/8.48 proper(true) -> ok(true) 26.78/8.48 proper(false) -> ok(false) 26.78/8.48 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 26.78/8.48 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 26.78/8.48 proper(0) -> ok(0) 26.78/8.48 proper(s(X)) -> s(proper(X)) 26.78/8.48 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 26.78/8.48 proper(nil) -> ok(nil) 26.78/8.48 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 26.78/8.48 proper(from(X)) -> from(proper(X)) 26.78/8.48 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 26.78/8.48 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 26.78/8.48 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 26.78/8.48 s(ok(X)) -> ok(s(X)) 26.78/8.48 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 26.78/8.48 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 26.78/8.48 from(ok(X)) -> ok(from(X)) 26.78/8.48 top(mark(X)) -> top(proper(X)) 26.78/8.48 top(ok(X)) -> top(active(X)) 26.78/8.48 26.78/8.48 S is empty. 26.78/8.48 Rewrite Strategy: FULL 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 26.78/8.48 The following defined symbols can occur below the 0th argument of top: proper, active 26.78/8.48 The following defined symbols can occur below the 0th argument of proper: proper, active 26.78/8.48 The following defined symbols can occur below the 0th argument of active: proper, active 26.78/8.48 26.78/8.48 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 26.78/8.48 active(and(true, X)) -> mark(X) 26.78/8.48 active(and(false, Y)) -> mark(false) 26.78/8.48 active(if(true, X, Y)) -> mark(X) 26.78/8.48 active(if(false, X, Y)) -> mark(Y) 26.78/8.48 active(add(0, X)) -> mark(X) 26.78/8.48 active(add(s(X), Y)) -> mark(s(add(X, Y))) 26.78/8.48 active(first(0, X)) -> mark(nil) 26.78/8.48 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 26.78/8.48 active(from(X)) -> mark(cons(X, from(s(X)))) 26.78/8.48 active(and(X1, X2)) -> and(active(X1), X2) 26.78/8.48 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 26.78/8.48 active(add(X1, X2)) -> add(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(X1, active(X2)) 26.78/8.48 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 26.78/8.48 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 26.78/8.48 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 26.78/8.48 proper(s(X)) -> s(proper(X)) 26.78/8.48 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 26.78/8.48 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 26.78/8.48 proper(from(X)) -> from(proper(X)) 26.78/8.48 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (2) 26.78/8.48 Obligation: 26.78/8.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 26.78/8.48 26.78/8.48 26.78/8.48 The TRS R consists of the following rules: 26.78/8.48 26.78/8.48 and(mark(X1), X2) -> mark(and(X1, X2)) 26.78/8.48 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 26.78/8.48 add(mark(X1), X2) -> mark(add(X1, X2)) 26.78/8.48 first(mark(X1), X2) -> mark(first(X1, X2)) 26.78/8.48 first(X1, mark(X2)) -> mark(first(X1, X2)) 26.78/8.48 proper(true) -> ok(true) 26.78/8.48 proper(false) -> ok(false) 26.78/8.48 proper(0) -> ok(0) 26.78/8.48 proper(nil) -> ok(nil) 26.78/8.48 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 26.78/8.48 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 26.78/8.48 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 26.78/8.48 s(ok(X)) -> ok(s(X)) 26.78/8.48 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 26.78/8.48 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 26.78/8.48 from(ok(X)) -> ok(from(X)) 26.78/8.48 top(mark(X)) -> top(proper(X)) 26.78/8.48 top(ok(X)) -> top(active(X)) 26.78/8.48 26.78/8.48 S is empty. 26.78/8.48 Rewrite Strategy: FULL 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 26.78/8.48 transformed relative TRS to TRS 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (4) 26.78/8.48 Obligation: 26.78/8.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 26.78/8.48 26.78/8.48 26.78/8.48 The TRS R consists of the following rules: 26.78/8.48 26.78/8.48 and(mark(X1), X2) -> mark(and(X1, X2)) 26.78/8.48 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 26.78/8.48 add(mark(X1), X2) -> mark(add(X1, X2)) 26.78/8.48 first(mark(X1), X2) -> mark(first(X1, X2)) 26.78/8.48 first(X1, mark(X2)) -> mark(first(X1, X2)) 26.78/8.48 proper(true) -> ok(true) 26.78/8.48 proper(false) -> ok(false) 26.78/8.48 proper(0) -> ok(0) 26.78/8.48 proper(nil) -> ok(nil) 26.78/8.48 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 26.78/8.48 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 26.78/8.48 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 26.78/8.48 s(ok(X)) -> ok(s(X)) 26.78/8.48 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 26.78/8.48 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 26.78/8.48 from(ok(X)) -> ok(from(X)) 26.78/8.48 top(mark(X)) -> top(proper(X)) 26.78/8.48 top(ok(X)) -> top(active(X)) 26.78/8.48 26.78/8.48 S is empty. 26.78/8.48 Rewrite Strategy: FULL 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (5) CpxTrsMatchBoundsTAProof (FINISHED) 26.78/8.48 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. 26.78/8.48 26.78/8.48 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 26.78/8.48 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9] 26.78/8.48 transitions: 26.78/8.48 mark0(0) -> 0 26.78/8.48 true0() -> 0 26.78/8.48 ok0(0) -> 0 26.78/8.48 false0() -> 0 26.78/8.48 00() -> 0 26.78/8.48 nil0() -> 0 26.78/8.48 active0(0) -> 0 26.78/8.48 and0(0, 0) -> 1 26.78/8.48 if0(0, 0, 0) -> 2 26.78/8.48 add0(0, 0) -> 3 26.78/8.48 first0(0, 0) -> 4 26.78/8.48 proper0(0) -> 5 26.78/8.48 s0(0) -> 6 26.78/8.48 cons0(0, 0) -> 7 26.78/8.48 from0(0) -> 8 26.78/8.48 top0(0) -> 9 26.78/8.48 and1(0, 0) -> 10 26.78/8.48 mark1(10) -> 1 26.78/8.48 if1(0, 0, 0) -> 11 26.78/8.48 mark1(11) -> 2 26.78/8.48 add1(0, 0) -> 12 26.78/8.48 mark1(12) -> 3 26.78/8.48 first1(0, 0) -> 13 26.78/8.48 mark1(13) -> 4 26.78/8.48 true1() -> 14 26.78/8.48 ok1(14) -> 5 26.78/8.48 false1() -> 15 26.78/8.48 ok1(15) -> 5 26.78/8.48 01() -> 16 26.78/8.48 ok1(16) -> 5 26.78/8.48 nil1() -> 17 26.78/8.48 ok1(17) -> 5 26.78/8.48 and1(0, 0) -> 18 26.78/8.48 ok1(18) -> 1 26.78/8.48 if1(0, 0, 0) -> 19 26.78/8.48 ok1(19) -> 2 26.78/8.48 add1(0, 0) -> 20 26.78/8.48 ok1(20) -> 3 26.78/8.48 s1(0) -> 21 26.78/8.48 ok1(21) -> 6 26.78/8.48 first1(0, 0) -> 22 26.78/8.48 ok1(22) -> 4 26.78/8.48 cons1(0, 0) -> 23 26.78/8.48 ok1(23) -> 7 26.78/8.48 from1(0) -> 24 26.78/8.48 ok1(24) -> 8 26.78/8.48 proper1(0) -> 25 26.78/8.48 top1(25) -> 9 26.78/8.48 active1(0) -> 26 26.78/8.48 top1(26) -> 9 26.78/8.48 mark1(10) -> 10 26.78/8.48 mark1(10) -> 18 26.78/8.48 mark1(11) -> 11 26.78/8.48 mark1(11) -> 19 26.78/8.48 mark1(12) -> 12 26.78/8.48 mark1(12) -> 20 26.78/8.48 mark1(13) -> 13 26.78/8.48 mark1(13) -> 22 26.78/8.48 ok1(14) -> 25 26.78/8.48 ok1(15) -> 25 26.78/8.48 ok1(16) -> 25 26.78/8.48 ok1(17) -> 25 26.78/8.48 ok1(18) -> 10 26.78/8.48 ok1(18) -> 18 26.78/8.48 ok1(19) -> 11 26.78/8.48 ok1(19) -> 19 26.78/8.48 ok1(20) -> 12 26.78/8.48 ok1(20) -> 20 26.78/8.48 ok1(21) -> 21 26.78/8.48 ok1(22) -> 13 26.78/8.48 ok1(22) -> 22 26.78/8.48 ok1(23) -> 23 26.78/8.48 ok1(24) -> 24 26.78/8.48 active2(14) -> 27 26.78/8.48 top2(27) -> 9 26.78/8.48 active2(15) -> 27 26.78/8.48 active2(16) -> 27 26.78/8.48 active2(17) -> 27 26.78/8.48 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (6) 26.78/8.48 BOUNDS(1, n^1) 26.78/8.48 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 26.78/8.48 Transformed a relative TRS into a decreasing-loop problem. 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (8) 26.78/8.48 Obligation: 26.78/8.48 Analyzing the following TRS for decreasing loops: 26.78/8.48 26.78/8.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 26.78/8.48 26.78/8.48 26.78/8.48 The TRS R consists of the following rules: 26.78/8.48 26.78/8.48 active(and(true, X)) -> mark(X) 26.78/8.48 active(and(false, Y)) -> mark(false) 26.78/8.48 active(if(true, X, Y)) -> mark(X) 26.78/8.48 active(if(false, X, Y)) -> mark(Y) 26.78/8.48 active(add(0, X)) -> mark(X) 26.78/8.48 active(add(s(X), Y)) -> mark(s(add(X, Y))) 26.78/8.48 active(first(0, X)) -> mark(nil) 26.78/8.48 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 26.78/8.48 active(from(X)) -> mark(cons(X, from(s(X)))) 26.78/8.48 active(and(X1, X2)) -> and(active(X1), X2) 26.78/8.48 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 26.78/8.48 active(add(X1, X2)) -> add(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(X1, active(X2)) 26.78/8.48 and(mark(X1), X2) -> mark(and(X1, X2)) 26.78/8.48 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 26.78/8.48 add(mark(X1), X2) -> mark(add(X1, X2)) 26.78/8.48 first(mark(X1), X2) -> mark(first(X1, X2)) 26.78/8.48 first(X1, mark(X2)) -> mark(first(X1, X2)) 26.78/8.48 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 26.78/8.48 proper(true) -> ok(true) 26.78/8.48 proper(false) -> ok(false) 26.78/8.48 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 26.78/8.48 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 26.78/8.48 proper(0) -> ok(0) 26.78/8.48 proper(s(X)) -> s(proper(X)) 26.78/8.48 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 26.78/8.48 proper(nil) -> ok(nil) 26.78/8.48 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 26.78/8.48 proper(from(X)) -> from(proper(X)) 26.78/8.48 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 26.78/8.48 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 26.78/8.48 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 26.78/8.48 s(ok(X)) -> ok(s(X)) 26.78/8.48 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 26.78/8.48 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 26.78/8.48 from(ok(X)) -> ok(from(X)) 26.78/8.48 top(mark(X)) -> top(proper(X)) 26.78/8.48 top(ok(X)) -> top(active(X)) 26.78/8.48 26.78/8.48 S is empty. 26.78/8.48 Rewrite Strategy: FULL 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (9) DecreasingLoopProof (LOWER BOUND(ID)) 26.78/8.48 The following loop(s) give(s) rise to the lower bound Omega(n^1): 26.78/8.48 26.78/8.48 The rewrite sequence 26.78/8.48 26.78/8.48 from(ok(X)) ->^+ ok(from(X)) 26.78/8.48 26.78/8.48 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 26.78/8.48 26.78/8.48 The pumping substitution is [X / ok(X)]. 26.78/8.48 26.78/8.48 The result substitution is [ ]. 26.78/8.48 26.78/8.48 26.78/8.48 26.78/8.48 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (10) 26.78/8.48 Complex Obligation (BEST) 26.78/8.48 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (11) 26.78/8.48 Obligation: 26.78/8.48 Proved the lower bound n^1 for the following obligation: 26.78/8.48 26.78/8.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 26.78/8.48 26.78/8.48 26.78/8.48 The TRS R consists of the following rules: 26.78/8.48 26.78/8.48 active(and(true, X)) -> mark(X) 26.78/8.48 active(and(false, Y)) -> mark(false) 26.78/8.48 active(if(true, X, Y)) -> mark(X) 26.78/8.48 active(if(false, X, Y)) -> mark(Y) 26.78/8.48 active(add(0, X)) -> mark(X) 26.78/8.48 active(add(s(X), Y)) -> mark(s(add(X, Y))) 26.78/8.48 active(first(0, X)) -> mark(nil) 26.78/8.48 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 26.78/8.48 active(from(X)) -> mark(cons(X, from(s(X)))) 26.78/8.48 active(and(X1, X2)) -> and(active(X1), X2) 26.78/8.48 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 26.78/8.48 active(add(X1, X2)) -> add(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(X1, active(X2)) 26.78/8.48 and(mark(X1), X2) -> mark(and(X1, X2)) 26.78/8.48 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 26.78/8.48 add(mark(X1), X2) -> mark(add(X1, X2)) 26.78/8.48 first(mark(X1), X2) -> mark(first(X1, X2)) 26.78/8.48 first(X1, mark(X2)) -> mark(first(X1, X2)) 26.78/8.48 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 26.78/8.48 proper(true) -> ok(true) 26.78/8.48 proper(false) -> ok(false) 26.78/8.48 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 26.78/8.48 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 26.78/8.48 proper(0) -> ok(0) 26.78/8.48 proper(s(X)) -> s(proper(X)) 26.78/8.48 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 26.78/8.48 proper(nil) -> ok(nil) 26.78/8.48 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 26.78/8.48 proper(from(X)) -> from(proper(X)) 26.78/8.48 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 26.78/8.48 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 26.78/8.48 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 26.78/8.48 s(ok(X)) -> ok(s(X)) 26.78/8.48 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 26.78/8.48 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 26.78/8.48 from(ok(X)) -> ok(from(X)) 26.78/8.48 top(mark(X)) -> top(proper(X)) 26.78/8.48 top(ok(X)) -> top(active(X)) 26.78/8.48 26.78/8.48 S is empty. 26.78/8.48 Rewrite Strategy: FULL 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (12) LowerBoundPropagationProof (FINISHED) 26.78/8.48 Propagated lower bound. 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (13) 26.78/8.48 BOUNDS(n^1, INF) 26.78/8.48 26.78/8.48 ---------------------------------------- 26.78/8.48 26.78/8.48 (14) 26.78/8.48 Obligation: 26.78/8.48 Analyzing the following TRS for decreasing loops: 26.78/8.48 26.78/8.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 26.78/8.48 26.78/8.48 26.78/8.48 The TRS R consists of the following rules: 26.78/8.48 26.78/8.48 active(and(true, X)) -> mark(X) 26.78/8.48 active(and(false, Y)) -> mark(false) 26.78/8.48 active(if(true, X, Y)) -> mark(X) 26.78/8.48 active(if(false, X, Y)) -> mark(Y) 26.78/8.48 active(add(0, X)) -> mark(X) 26.78/8.48 active(add(s(X), Y)) -> mark(s(add(X, Y))) 26.78/8.48 active(first(0, X)) -> mark(nil) 26.78/8.48 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 26.78/8.48 active(from(X)) -> mark(cons(X, from(s(X)))) 26.78/8.48 active(and(X1, X2)) -> and(active(X1), X2) 26.78/8.48 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 26.78/8.48 active(add(X1, X2)) -> add(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(active(X1), X2) 26.78/8.48 active(first(X1, X2)) -> first(X1, active(X2)) 26.78/8.48 and(mark(X1), X2) -> mark(and(X1, X2)) 26.78/8.48 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 26.78/8.48 add(mark(X1), X2) -> mark(add(X1, X2)) 26.78/8.48 first(mark(X1), X2) -> mark(first(X1, X2)) 26.78/8.48 first(X1, mark(X2)) -> mark(first(X1, X2)) 26.78/8.48 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 26.78/8.48 proper(true) -> ok(true) 26.78/8.48 proper(false) -> ok(false) 26.78/8.48 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 26.78/8.48 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 26.78/8.48 proper(0) -> ok(0) 26.78/8.48 proper(s(X)) -> s(proper(X)) 26.78/8.48 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 26.78/8.48 proper(nil) -> ok(nil) 26.78/8.48 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 26.78/8.48 proper(from(X)) -> from(proper(X)) 26.78/8.48 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 26.78/8.48 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 26.78/8.48 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 26.78/8.48 s(ok(X)) -> ok(s(X)) 26.78/8.48 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 26.78/8.48 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 26.78/8.48 from(ok(X)) -> ok(from(X)) 26.78/8.48 top(mark(X)) -> top(proper(X)) 26.78/8.48 top(ok(X)) -> top(active(X)) 26.78/8.48 26.78/8.48 S is empty. 26.78/8.48 Rewrite Strategy: FULL 27.06/10.15 EOF