24.24/7.08 WORST_CASE(Omega(n^1), O(n^1)) 24.24/7.10 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.24/7.10 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.24/7.10 24.24/7.10 24.24/7.10 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.24/7.10 24.24/7.10 (0) CpxTRS 24.24/7.10 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 13 ms] 24.24/7.10 (2) CpxTRS 24.24/7.10 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.24/7.10 (4) CpxTRS 24.24/7.10 (5) CpxTrsMatchBoundsTAProof [FINISHED, 116 ms] 24.24/7.10 (6) BOUNDS(1, n^1) 24.24/7.10 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 24.24/7.10 (8) TRS for Loop Detection 24.24/7.10 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 24.24/7.10 (10) BEST 24.24/7.10 (11) proven lower bound 24.24/7.10 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 24.24/7.10 (13) BOUNDS(n^1, INF) 24.24/7.10 (14) TRS for Loop Detection 24.24/7.10 24.24/7.10 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (0) 24.24/7.10 Obligation: 24.24/7.10 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.24/7.10 24.24/7.10 24.24/7.10 The TRS R consists of the following rules: 24.24/7.10 24.24/7.10 active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X))))) 24.24/7.10 active(add(0, X)) -> mark(X) 24.24/7.10 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.24/7.10 active(prod(0, X)) -> mark(0) 24.24/7.10 active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y))) 24.24/7.10 active(if(true, X, Y)) -> mark(X) 24.24/7.10 active(if(false, X, Y)) -> mark(Y) 24.24/7.10 active(zero(0)) -> mark(true) 24.24/7.10 active(zero(s(X))) -> mark(false) 24.24/7.10 active(p(s(X))) -> mark(X) 24.24/7.10 active(fact(X)) -> fact(active(X)) 24.24/7.10 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.24/7.10 active(zero(X)) -> zero(active(X)) 24.24/7.10 active(s(X)) -> s(active(X)) 24.24/7.10 active(prod(X1, X2)) -> prod(active(X1), X2) 24.24/7.10 active(prod(X1, X2)) -> prod(X1, active(X2)) 24.24/7.10 active(p(X)) -> p(active(X)) 24.24/7.10 active(add(X1, X2)) -> add(active(X1), X2) 24.24/7.10 active(add(X1, X2)) -> add(X1, active(X2)) 24.24/7.10 fact(mark(X)) -> mark(fact(X)) 24.24/7.10 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.24/7.10 zero(mark(X)) -> mark(zero(X)) 24.24/7.10 s(mark(X)) -> mark(s(X)) 24.24/7.10 prod(mark(X1), X2) -> mark(prod(X1, X2)) 24.24/7.10 prod(X1, mark(X2)) -> mark(prod(X1, X2)) 24.24/7.10 p(mark(X)) -> mark(p(X)) 24.24/7.10 add(mark(X1), X2) -> mark(add(X1, X2)) 24.24/7.10 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.24/7.10 proper(fact(X)) -> fact(proper(X)) 24.24/7.10 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.24/7.10 proper(zero(X)) -> zero(proper(X)) 24.24/7.10 proper(s(X)) -> s(proper(X)) 24.24/7.10 proper(0) -> ok(0) 24.24/7.10 proper(prod(X1, X2)) -> prod(proper(X1), proper(X2)) 24.24/7.10 proper(p(X)) -> p(proper(X)) 24.24/7.10 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.24/7.10 proper(true) -> ok(true) 24.24/7.10 proper(false) -> ok(false) 24.24/7.10 fact(ok(X)) -> ok(fact(X)) 24.24/7.10 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.24/7.10 zero(ok(X)) -> ok(zero(X)) 24.24/7.10 s(ok(X)) -> ok(s(X)) 24.24/7.10 prod(ok(X1), ok(X2)) -> ok(prod(X1, X2)) 24.24/7.10 p(ok(X)) -> ok(p(X)) 24.24/7.10 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.24/7.10 top(mark(X)) -> top(proper(X)) 24.24/7.10 top(ok(X)) -> top(active(X)) 24.24/7.10 24.24/7.10 S is empty. 24.24/7.10 Rewrite Strategy: FULL 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.24/7.10 The following defined symbols can occur below the 0th argument of top: proper, active 24.24/7.10 The following defined symbols can occur below the 0th argument of proper: proper, active 24.24/7.10 The following defined symbols can occur below the 0th argument of active: proper, active 24.24/7.10 24.24/7.10 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.24/7.10 active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X))))) 24.24/7.10 active(add(0, X)) -> mark(X) 24.24/7.10 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.24/7.10 active(prod(0, X)) -> mark(0) 24.24/7.10 active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y))) 24.24/7.10 active(if(true, X, Y)) -> mark(X) 24.24/7.10 active(if(false, X, Y)) -> mark(Y) 24.24/7.10 active(zero(0)) -> mark(true) 24.24/7.10 active(zero(s(X))) -> mark(false) 24.24/7.10 active(p(s(X))) -> mark(X) 24.24/7.10 active(fact(X)) -> fact(active(X)) 24.24/7.10 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.24/7.10 active(zero(X)) -> zero(active(X)) 24.24/7.10 active(s(X)) -> s(active(X)) 24.24/7.10 active(prod(X1, X2)) -> prod(active(X1), X2) 24.24/7.10 active(prod(X1, X2)) -> prod(X1, active(X2)) 24.24/7.10 active(p(X)) -> p(active(X)) 24.24/7.10 active(add(X1, X2)) -> add(active(X1), X2) 24.24/7.10 active(add(X1, X2)) -> add(X1, active(X2)) 24.24/7.10 proper(fact(X)) -> fact(proper(X)) 24.24/7.10 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.24/7.10 proper(zero(X)) -> zero(proper(X)) 24.24/7.10 proper(s(X)) -> s(proper(X)) 24.24/7.10 proper(prod(X1, X2)) -> prod(proper(X1), proper(X2)) 24.24/7.10 proper(p(X)) -> p(proper(X)) 24.24/7.10 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.24/7.10 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (2) 24.24/7.10 Obligation: 24.24/7.10 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.24/7.10 24.24/7.10 24.24/7.10 The TRS R consists of the following rules: 24.24/7.10 24.24/7.10 fact(mark(X)) -> mark(fact(X)) 24.24/7.10 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.24/7.10 zero(mark(X)) -> mark(zero(X)) 24.24/7.10 s(mark(X)) -> mark(s(X)) 24.24/7.10 prod(mark(X1), X2) -> mark(prod(X1, X2)) 24.24/7.10 prod(X1, mark(X2)) -> mark(prod(X1, X2)) 24.24/7.10 p(mark(X)) -> mark(p(X)) 24.24/7.10 add(mark(X1), X2) -> mark(add(X1, X2)) 24.24/7.10 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.24/7.10 proper(0) -> ok(0) 24.24/7.10 proper(true) -> ok(true) 24.24/7.10 proper(false) -> ok(false) 24.24/7.10 fact(ok(X)) -> ok(fact(X)) 24.24/7.10 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.24/7.10 zero(ok(X)) -> ok(zero(X)) 24.24/7.10 s(ok(X)) -> ok(s(X)) 24.24/7.10 prod(ok(X1), ok(X2)) -> ok(prod(X1, X2)) 24.24/7.10 p(ok(X)) -> ok(p(X)) 24.24/7.10 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.24/7.10 top(mark(X)) -> top(proper(X)) 24.24/7.10 top(ok(X)) -> top(active(X)) 24.24/7.10 24.24/7.10 S is empty. 24.24/7.10 Rewrite Strategy: FULL 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.24/7.10 transformed relative TRS to TRS 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (4) 24.24/7.10 Obligation: 24.24/7.10 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.24/7.10 24.24/7.10 24.24/7.10 The TRS R consists of the following rules: 24.24/7.10 24.24/7.10 fact(mark(X)) -> mark(fact(X)) 24.24/7.10 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.24/7.10 zero(mark(X)) -> mark(zero(X)) 24.24/7.10 s(mark(X)) -> mark(s(X)) 24.24/7.10 prod(mark(X1), X2) -> mark(prod(X1, X2)) 24.24/7.10 prod(X1, mark(X2)) -> mark(prod(X1, X2)) 24.24/7.10 p(mark(X)) -> mark(p(X)) 24.24/7.10 add(mark(X1), X2) -> mark(add(X1, X2)) 24.24/7.10 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.24/7.10 proper(0) -> ok(0) 24.24/7.10 proper(true) -> ok(true) 24.24/7.10 proper(false) -> ok(false) 24.24/7.10 fact(ok(X)) -> ok(fact(X)) 24.24/7.10 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.24/7.10 zero(ok(X)) -> ok(zero(X)) 24.24/7.10 s(ok(X)) -> ok(s(X)) 24.24/7.10 prod(ok(X1), ok(X2)) -> ok(prod(X1, X2)) 24.24/7.10 p(ok(X)) -> ok(p(X)) 24.24/7.10 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.24/7.10 top(mark(X)) -> top(proper(X)) 24.24/7.10 top(ok(X)) -> top(active(X)) 24.24/7.10 24.24/7.10 S is empty. 24.24/7.10 Rewrite Strategy: FULL 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.24/7.10 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. 24.24/7.10 24.24/7.10 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.24/7.10 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9] 24.24/7.10 transitions: 24.24/7.10 mark0(0) -> 0 24.24/7.10 00() -> 0 24.24/7.10 ok0(0) -> 0 24.24/7.10 true0() -> 0 24.24/7.10 false0() -> 0 24.24/7.10 active0(0) -> 0 24.24/7.10 fact0(0) -> 1 24.24/7.10 if0(0, 0, 0) -> 2 24.24/7.10 zero0(0) -> 3 24.24/7.10 s0(0) -> 4 24.24/7.10 prod0(0, 0) -> 5 24.24/7.10 p0(0) -> 6 24.24/7.10 add0(0, 0) -> 7 24.24/7.10 proper0(0) -> 8 24.24/7.10 top0(0) -> 9 24.24/7.10 fact1(0) -> 10 24.24/7.10 mark1(10) -> 1 24.24/7.10 if1(0, 0, 0) -> 11 24.24/7.10 mark1(11) -> 2 24.24/7.10 zero1(0) -> 12 24.24/7.10 mark1(12) -> 3 24.24/7.10 s1(0) -> 13 24.24/7.10 mark1(13) -> 4 24.24/7.10 prod1(0, 0) -> 14 24.24/7.10 mark1(14) -> 5 24.24/7.10 p1(0) -> 15 24.24/7.10 mark1(15) -> 6 24.24/7.10 add1(0, 0) -> 16 24.24/7.10 mark1(16) -> 7 24.24/7.10 01() -> 17 24.24/7.10 ok1(17) -> 8 24.24/7.10 true1() -> 18 24.24/7.10 ok1(18) -> 8 24.24/7.10 false1() -> 19 24.24/7.10 ok1(19) -> 8 24.24/7.10 fact1(0) -> 20 24.24/7.10 ok1(20) -> 1 24.24/7.10 if1(0, 0, 0) -> 21 24.24/7.10 ok1(21) -> 2 24.24/7.10 zero1(0) -> 22 24.24/7.10 ok1(22) -> 3 24.24/7.10 s1(0) -> 23 24.24/7.10 ok1(23) -> 4 24.24/7.10 prod1(0, 0) -> 24 24.24/7.10 ok1(24) -> 5 24.24/7.10 p1(0) -> 25 24.24/7.10 ok1(25) -> 6 24.24/7.10 add1(0, 0) -> 26 24.24/7.10 ok1(26) -> 7 24.24/7.10 proper1(0) -> 27 24.24/7.10 top1(27) -> 9 24.24/7.10 active1(0) -> 28 24.24/7.10 top1(28) -> 9 24.24/7.10 mark1(10) -> 10 24.24/7.10 mark1(10) -> 20 24.24/7.10 mark1(11) -> 11 24.24/7.10 mark1(11) -> 21 24.24/7.10 mark1(12) -> 12 24.24/7.10 mark1(12) -> 22 24.24/7.10 mark1(13) -> 13 24.24/7.10 mark1(13) -> 23 24.24/7.10 mark1(14) -> 14 24.24/7.10 mark1(14) -> 24 24.24/7.10 mark1(15) -> 15 24.24/7.10 mark1(15) -> 25 24.24/7.10 mark1(16) -> 16 24.24/7.10 mark1(16) -> 26 24.24/7.10 ok1(17) -> 27 24.24/7.10 ok1(18) -> 27 24.24/7.10 ok1(19) -> 27 24.24/7.10 ok1(20) -> 10 24.24/7.10 ok1(20) -> 20 24.24/7.10 ok1(21) -> 11 24.24/7.10 ok1(21) -> 21 24.24/7.10 ok1(22) -> 12 24.24/7.10 ok1(22) -> 22 24.24/7.10 ok1(23) -> 13 24.24/7.10 ok1(23) -> 23 24.24/7.10 ok1(24) -> 14 24.24/7.10 ok1(24) -> 24 24.24/7.10 ok1(25) -> 15 24.24/7.10 ok1(25) -> 25 24.24/7.10 ok1(26) -> 16 24.24/7.10 ok1(26) -> 26 24.24/7.10 active2(17) -> 29 24.24/7.10 top2(29) -> 9 24.24/7.10 active2(18) -> 29 24.24/7.10 active2(19) -> 29 24.24/7.10 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (6) 24.24/7.10 BOUNDS(1, n^1) 24.24/7.10 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 24.24/7.10 Transformed a relative TRS into a decreasing-loop problem. 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (8) 24.24/7.10 Obligation: 24.24/7.10 Analyzing the following TRS for decreasing loops: 24.24/7.10 24.24/7.10 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.24/7.10 24.24/7.10 24.24/7.10 The TRS R consists of the following rules: 24.24/7.10 24.24/7.10 active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X))))) 24.24/7.10 active(add(0, X)) -> mark(X) 24.24/7.10 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.24/7.10 active(prod(0, X)) -> mark(0) 24.24/7.10 active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y))) 24.24/7.10 active(if(true, X, Y)) -> mark(X) 24.24/7.10 active(if(false, X, Y)) -> mark(Y) 24.24/7.10 active(zero(0)) -> mark(true) 24.24/7.10 active(zero(s(X))) -> mark(false) 24.24/7.10 active(p(s(X))) -> mark(X) 24.24/7.10 active(fact(X)) -> fact(active(X)) 24.24/7.10 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.24/7.10 active(zero(X)) -> zero(active(X)) 24.24/7.10 active(s(X)) -> s(active(X)) 24.24/7.10 active(prod(X1, X2)) -> prod(active(X1), X2) 24.24/7.10 active(prod(X1, X2)) -> prod(X1, active(X2)) 24.24/7.10 active(p(X)) -> p(active(X)) 24.24/7.10 active(add(X1, X2)) -> add(active(X1), X2) 24.24/7.10 active(add(X1, X2)) -> add(X1, active(X2)) 24.24/7.10 fact(mark(X)) -> mark(fact(X)) 24.24/7.10 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.24/7.10 zero(mark(X)) -> mark(zero(X)) 24.24/7.10 s(mark(X)) -> mark(s(X)) 24.24/7.10 prod(mark(X1), X2) -> mark(prod(X1, X2)) 24.24/7.10 prod(X1, mark(X2)) -> mark(prod(X1, X2)) 24.24/7.10 p(mark(X)) -> mark(p(X)) 24.24/7.10 add(mark(X1), X2) -> mark(add(X1, X2)) 24.24/7.10 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.24/7.10 proper(fact(X)) -> fact(proper(X)) 24.24/7.10 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.24/7.10 proper(zero(X)) -> zero(proper(X)) 24.24/7.10 proper(s(X)) -> s(proper(X)) 24.24/7.10 proper(0) -> ok(0) 24.24/7.10 proper(prod(X1, X2)) -> prod(proper(X1), proper(X2)) 24.24/7.10 proper(p(X)) -> p(proper(X)) 24.24/7.10 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.24/7.10 proper(true) -> ok(true) 24.24/7.10 proper(false) -> ok(false) 24.24/7.10 fact(ok(X)) -> ok(fact(X)) 24.24/7.10 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.24/7.10 zero(ok(X)) -> ok(zero(X)) 24.24/7.10 s(ok(X)) -> ok(s(X)) 24.24/7.10 prod(ok(X1), ok(X2)) -> ok(prod(X1, X2)) 24.24/7.10 p(ok(X)) -> ok(p(X)) 24.24/7.10 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.24/7.10 top(mark(X)) -> top(proper(X)) 24.24/7.10 top(ok(X)) -> top(active(X)) 24.24/7.10 24.24/7.10 S is empty. 24.24/7.10 Rewrite Strategy: FULL 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (9) DecreasingLoopProof (LOWER BOUND(ID)) 24.24/7.10 The following loop(s) give(s) rise to the lower bound Omega(n^1): 24.24/7.10 24.24/7.10 The rewrite sequence 24.24/7.10 24.24/7.10 p(mark(X)) ->^+ mark(p(X)) 24.24/7.10 24.24/7.10 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 24.24/7.10 24.24/7.10 The pumping substitution is [X / mark(X)]. 24.24/7.10 24.24/7.10 The result substitution is [ ]. 24.24/7.10 24.24/7.10 24.24/7.10 24.24/7.10 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (10) 24.24/7.10 Complex Obligation (BEST) 24.24/7.10 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (11) 24.24/7.10 Obligation: 24.24/7.10 Proved the lower bound n^1 for the following obligation: 24.24/7.10 24.24/7.10 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.24/7.10 24.24/7.10 24.24/7.10 The TRS R consists of the following rules: 24.24/7.10 24.24/7.10 active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X))))) 24.24/7.10 active(add(0, X)) -> mark(X) 24.24/7.10 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.24/7.10 active(prod(0, X)) -> mark(0) 24.24/7.10 active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y))) 24.24/7.10 active(if(true, X, Y)) -> mark(X) 24.24/7.10 active(if(false, X, Y)) -> mark(Y) 24.24/7.10 active(zero(0)) -> mark(true) 24.24/7.10 active(zero(s(X))) -> mark(false) 24.24/7.10 active(p(s(X))) -> mark(X) 24.24/7.10 active(fact(X)) -> fact(active(X)) 24.24/7.10 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.24/7.10 active(zero(X)) -> zero(active(X)) 24.24/7.10 active(s(X)) -> s(active(X)) 24.24/7.10 active(prod(X1, X2)) -> prod(active(X1), X2) 24.24/7.10 active(prod(X1, X2)) -> prod(X1, active(X2)) 24.24/7.10 active(p(X)) -> p(active(X)) 24.24/7.10 active(add(X1, X2)) -> add(active(X1), X2) 24.24/7.10 active(add(X1, X2)) -> add(X1, active(X2)) 24.24/7.10 fact(mark(X)) -> mark(fact(X)) 24.24/7.10 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.24/7.10 zero(mark(X)) -> mark(zero(X)) 24.24/7.10 s(mark(X)) -> mark(s(X)) 24.24/7.10 prod(mark(X1), X2) -> mark(prod(X1, X2)) 24.24/7.10 prod(X1, mark(X2)) -> mark(prod(X1, X2)) 24.24/7.10 p(mark(X)) -> mark(p(X)) 24.24/7.10 add(mark(X1), X2) -> mark(add(X1, X2)) 24.24/7.10 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.24/7.10 proper(fact(X)) -> fact(proper(X)) 24.24/7.10 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.24/7.10 proper(zero(X)) -> zero(proper(X)) 24.24/7.10 proper(s(X)) -> s(proper(X)) 24.24/7.10 proper(0) -> ok(0) 24.24/7.10 proper(prod(X1, X2)) -> prod(proper(X1), proper(X2)) 24.24/7.10 proper(p(X)) -> p(proper(X)) 24.24/7.10 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.24/7.10 proper(true) -> ok(true) 24.24/7.10 proper(false) -> ok(false) 24.24/7.10 fact(ok(X)) -> ok(fact(X)) 24.24/7.10 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.24/7.10 zero(ok(X)) -> ok(zero(X)) 24.24/7.10 s(ok(X)) -> ok(s(X)) 24.24/7.10 prod(ok(X1), ok(X2)) -> ok(prod(X1, X2)) 24.24/7.10 p(ok(X)) -> ok(p(X)) 24.24/7.10 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.24/7.10 top(mark(X)) -> top(proper(X)) 24.24/7.10 top(ok(X)) -> top(active(X)) 24.24/7.10 24.24/7.10 S is empty. 24.24/7.10 Rewrite Strategy: FULL 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (12) LowerBoundPropagationProof (FINISHED) 24.24/7.10 Propagated lower bound. 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (13) 24.24/7.10 BOUNDS(n^1, INF) 24.24/7.10 24.24/7.10 ---------------------------------------- 24.24/7.10 24.24/7.10 (14) 24.24/7.10 Obligation: 24.24/7.10 Analyzing the following TRS for decreasing loops: 24.24/7.10 24.24/7.10 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.24/7.10 24.24/7.10 24.24/7.10 The TRS R consists of the following rules: 24.24/7.10 24.24/7.10 active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X))))) 24.24/7.10 active(add(0, X)) -> mark(X) 24.24/7.10 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.24/7.10 active(prod(0, X)) -> mark(0) 24.24/7.10 active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y))) 24.24/7.10 active(if(true, X, Y)) -> mark(X) 24.24/7.10 active(if(false, X, Y)) -> mark(Y) 24.24/7.10 active(zero(0)) -> mark(true) 24.24/7.10 active(zero(s(X))) -> mark(false) 24.24/7.10 active(p(s(X))) -> mark(X) 24.24/7.10 active(fact(X)) -> fact(active(X)) 24.24/7.10 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.24/7.10 active(zero(X)) -> zero(active(X)) 24.24/7.10 active(s(X)) -> s(active(X)) 24.24/7.10 active(prod(X1, X2)) -> prod(active(X1), X2) 24.24/7.10 active(prod(X1, X2)) -> prod(X1, active(X2)) 24.24/7.10 active(p(X)) -> p(active(X)) 24.24/7.10 active(add(X1, X2)) -> add(active(X1), X2) 24.24/7.10 active(add(X1, X2)) -> add(X1, active(X2)) 24.24/7.10 fact(mark(X)) -> mark(fact(X)) 24.24/7.10 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.24/7.10 zero(mark(X)) -> mark(zero(X)) 24.24/7.10 s(mark(X)) -> mark(s(X)) 24.24/7.10 prod(mark(X1), X2) -> mark(prod(X1, X2)) 24.24/7.10 prod(X1, mark(X2)) -> mark(prod(X1, X2)) 24.24/7.10 p(mark(X)) -> mark(p(X)) 24.24/7.10 add(mark(X1), X2) -> mark(add(X1, X2)) 24.24/7.10 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.24/7.10 proper(fact(X)) -> fact(proper(X)) 24.24/7.10 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.24/7.10 proper(zero(X)) -> zero(proper(X)) 24.24/7.10 proper(s(X)) -> s(proper(X)) 24.24/7.10 proper(0) -> ok(0) 24.24/7.10 proper(prod(X1, X2)) -> prod(proper(X1), proper(X2)) 24.24/7.10 proper(p(X)) -> p(proper(X)) 24.24/7.10 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.24/7.10 proper(true) -> ok(true) 24.24/7.10 proper(false) -> ok(false) 24.24/7.10 fact(ok(X)) -> ok(fact(X)) 24.24/7.10 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.24/7.10 zero(ok(X)) -> ok(zero(X)) 24.24/7.10 s(ok(X)) -> ok(s(X)) 24.24/7.10 prod(ok(X1), ok(X2)) -> ok(prod(X1, X2)) 24.24/7.10 p(ok(X)) -> ok(p(X)) 24.24/7.10 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.24/7.10 top(mark(X)) -> top(proper(X)) 24.24/7.10 top(ok(X)) -> top(active(X)) 24.24/7.10 24.24/7.10 S is empty. 24.24/7.10 Rewrite Strategy: FULL 24.58/7.15 EOF