1089.44/291.52 WORST_CASE(Omega(n^1), ?) 1089.44/291.54 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1089.44/291.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1089.44/291.54 1089.44/291.54 1089.44/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1089.44/291.54 1089.44/291.54 (0) CpxTRS 1089.44/291.54 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1089.44/291.54 (2) TRS for Loop Detection 1089.44/291.54 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1089.44/291.54 (4) BEST 1089.44/291.54 (5) proven lower bound 1089.44/291.54 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1089.44/291.54 (7) BOUNDS(n^1, INF) 1089.44/291.54 (8) TRS for Loop Detection 1089.44/291.54 1089.44/291.54 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (0) 1089.44/291.54 Obligation: 1089.44/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1089.44/291.54 1089.44/291.54 1089.44/291.54 The TRS R consists of the following rules: 1089.44/291.54 1089.44/291.54 eq(0, 0) -> true 1089.44/291.54 eq(0, s(x)) -> false 1089.44/291.54 eq(s(x), 0) -> false 1089.44/291.54 eq(s(x), s(y)) -> eq(x, y) 1089.44/291.54 or(true, y) -> true 1089.44/291.54 or(false, y) -> y 1089.44/291.54 union(empty, h) -> h 1089.44/291.54 union(edge(x, y, i), h) -> edge(x, y, union(i, h)) 1089.44/291.54 reach(x, y, empty, h) -> false 1089.44/291.54 reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h) 1089.44/291.54 if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h) 1089.44/291.54 if_reach_2(true, x, y, edge(u, v, i), h) -> true 1089.44/291.54 if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty)) 1089.44/291.54 if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h)) 1089.44/291.54 1089.44/291.54 S is empty. 1089.44/291.54 Rewrite Strategy: INNERMOST 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1089.44/291.54 Transformed a relative TRS into a decreasing-loop problem. 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (2) 1089.44/291.54 Obligation: 1089.44/291.54 Analyzing the following TRS for decreasing loops: 1089.44/291.54 1089.44/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1089.44/291.54 1089.44/291.54 1089.44/291.54 The TRS R consists of the following rules: 1089.44/291.54 1089.44/291.54 eq(0, 0) -> true 1089.44/291.54 eq(0, s(x)) -> false 1089.44/291.54 eq(s(x), 0) -> false 1089.44/291.54 eq(s(x), s(y)) -> eq(x, y) 1089.44/291.54 or(true, y) -> true 1089.44/291.54 or(false, y) -> y 1089.44/291.54 union(empty, h) -> h 1089.44/291.54 union(edge(x, y, i), h) -> edge(x, y, union(i, h)) 1089.44/291.54 reach(x, y, empty, h) -> false 1089.44/291.54 reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h) 1089.44/291.54 if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h) 1089.44/291.54 if_reach_2(true, x, y, edge(u, v, i), h) -> true 1089.44/291.54 if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty)) 1089.44/291.54 if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h)) 1089.44/291.54 1089.44/291.54 S is empty. 1089.44/291.54 Rewrite Strategy: INNERMOST 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1089.44/291.54 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1089.44/291.54 1089.44/291.54 The rewrite sequence 1089.44/291.54 1089.44/291.54 union(edge(x, y, i), h) ->^+ edge(x, y, union(i, h)) 1089.44/291.54 1089.44/291.54 gives rise to a decreasing loop by considering the right hand sides subterm at position [2]. 1089.44/291.54 1089.44/291.54 The pumping substitution is [i / edge(x, y, i)]. 1089.44/291.54 1089.44/291.54 The result substitution is [ ]. 1089.44/291.54 1089.44/291.54 1089.44/291.54 1089.44/291.54 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (4) 1089.44/291.54 Complex Obligation (BEST) 1089.44/291.54 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (5) 1089.44/291.54 Obligation: 1089.44/291.54 Proved the lower bound n^1 for the following obligation: 1089.44/291.54 1089.44/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1089.44/291.54 1089.44/291.54 1089.44/291.54 The TRS R consists of the following rules: 1089.44/291.54 1089.44/291.54 eq(0, 0) -> true 1089.44/291.54 eq(0, s(x)) -> false 1089.44/291.54 eq(s(x), 0) -> false 1089.44/291.54 eq(s(x), s(y)) -> eq(x, y) 1089.44/291.54 or(true, y) -> true 1089.44/291.54 or(false, y) -> y 1089.44/291.54 union(empty, h) -> h 1089.44/291.54 union(edge(x, y, i), h) -> edge(x, y, union(i, h)) 1089.44/291.54 reach(x, y, empty, h) -> false 1089.44/291.54 reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h) 1089.44/291.54 if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h) 1089.44/291.54 if_reach_2(true, x, y, edge(u, v, i), h) -> true 1089.44/291.54 if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty)) 1089.44/291.54 if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h)) 1089.44/291.54 1089.44/291.54 S is empty. 1089.44/291.54 Rewrite Strategy: INNERMOST 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (6) LowerBoundPropagationProof (FINISHED) 1089.44/291.54 Propagated lower bound. 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (7) 1089.44/291.54 BOUNDS(n^1, INF) 1089.44/291.54 1089.44/291.54 ---------------------------------------- 1089.44/291.54 1089.44/291.54 (8) 1089.44/291.54 Obligation: 1089.44/291.54 Analyzing the following TRS for decreasing loops: 1089.44/291.54 1089.44/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1089.44/291.54 1089.44/291.54 1089.44/291.54 The TRS R consists of the following rules: 1089.44/291.54 1089.44/291.54 eq(0, 0) -> true 1089.44/291.54 eq(0, s(x)) -> false 1089.44/291.54 eq(s(x), 0) -> false 1089.44/291.54 eq(s(x), s(y)) -> eq(x, y) 1089.44/291.54 or(true, y) -> true 1089.44/291.54 or(false, y) -> y 1089.44/291.54 union(empty, h) -> h 1089.44/291.54 union(edge(x, y, i), h) -> edge(x, y, union(i, h)) 1089.44/291.54 reach(x, y, empty, h) -> false 1089.44/291.54 reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h) 1089.44/291.54 if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h) 1089.44/291.54 if_reach_2(true, x, y, edge(u, v, i), h) -> true 1089.44/291.54 if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty)) 1089.44/291.54 if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h)) 1089.44/291.54 1089.44/291.54 S is empty. 1089.44/291.54 Rewrite Strategy: INNERMOST 1089.71/291.61 EOF