1141.28/291.50 WORST_CASE(Omega(n^1), ?) 1154.08/294.73 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1154.08/294.73 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1154.08/294.73 1154.08/294.73 1154.08/294.73 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1154.08/294.73 1154.08/294.73 (0) CpxTRS 1154.08/294.73 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1154.08/294.73 (2) TRS for Loop Detection 1154.08/294.73 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1154.08/294.73 (4) BEST 1154.08/294.73 (5) proven lower bound 1154.08/294.73 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1154.08/294.73 (7) BOUNDS(n^1, INF) 1154.08/294.73 (8) TRS for Loop Detection 1154.08/294.73 1154.08/294.73 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (0) 1154.08/294.73 Obligation: 1154.08/294.73 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1154.08/294.73 1154.08/294.73 1154.08/294.73 The TRS R consists of the following rules: 1154.08/294.73 1154.08/294.73 eq(0, 0) -> true 1154.08/294.73 eq(0, s(x)) -> false 1154.08/294.73 eq(s(x), 0) -> false 1154.08/294.73 eq(s(x), s(y)) -> eq(x, y) 1154.08/294.73 le(0, y) -> true 1154.08/294.73 le(s(x), 0) -> false 1154.08/294.73 le(s(x), s(y)) -> le(x, y) 1154.08/294.73 app(nil, y) -> y 1154.08/294.73 app(add(n, x), y) -> add(n, app(x, y)) 1154.08/294.73 min(add(n, nil)) -> n 1154.08/294.73 min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) 1154.08/294.73 if_min(true, add(n, add(m, x))) -> min(add(n, x)) 1154.08/294.73 if_min(false, add(n, add(m, x))) -> min(add(m, x)) 1154.08/294.73 head(add(n, x)) -> n 1154.08/294.73 tail(add(n, x)) -> x 1154.08/294.73 tail(nil) -> nil 1154.08/294.73 null(nil) -> true 1154.08/294.73 null(add(n, x)) -> false 1154.08/294.73 rm(n, nil) -> nil 1154.08/294.73 rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) 1154.08/294.73 if_rm(true, n, add(m, x)) -> rm(n, x) 1154.08/294.73 if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) 1154.08/294.73 minsort(x) -> mins(x, nil, nil) 1154.08/294.73 mins(x, y, z) -> if(null(x), x, y, z) 1154.08/294.73 if(true, x, y, z) -> z 1154.08/294.73 if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) 1154.08/294.73 if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) 1154.08/294.73 if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) 1154.08/294.73 1154.08/294.73 S is empty. 1154.08/294.73 Rewrite Strategy: INNERMOST 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1154.08/294.73 Transformed a relative TRS into a decreasing-loop problem. 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (2) 1154.08/294.73 Obligation: 1154.08/294.73 Analyzing the following TRS for decreasing loops: 1154.08/294.73 1154.08/294.73 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1154.08/294.73 1154.08/294.73 1154.08/294.73 The TRS R consists of the following rules: 1154.08/294.73 1154.08/294.73 eq(0, 0) -> true 1154.08/294.73 eq(0, s(x)) -> false 1154.08/294.73 eq(s(x), 0) -> false 1154.08/294.73 eq(s(x), s(y)) -> eq(x, y) 1154.08/294.73 le(0, y) -> true 1154.08/294.73 le(s(x), 0) -> false 1154.08/294.73 le(s(x), s(y)) -> le(x, y) 1154.08/294.73 app(nil, y) -> y 1154.08/294.73 app(add(n, x), y) -> add(n, app(x, y)) 1154.08/294.73 min(add(n, nil)) -> n 1154.08/294.73 min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) 1154.08/294.73 if_min(true, add(n, add(m, x))) -> min(add(n, x)) 1154.08/294.73 if_min(false, add(n, add(m, x))) -> min(add(m, x)) 1154.08/294.73 head(add(n, x)) -> n 1154.08/294.73 tail(add(n, x)) -> x 1154.08/294.73 tail(nil) -> nil 1154.08/294.73 null(nil) -> true 1154.08/294.73 null(add(n, x)) -> false 1154.08/294.73 rm(n, nil) -> nil 1154.08/294.73 rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) 1154.08/294.73 if_rm(true, n, add(m, x)) -> rm(n, x) 1154.08/294.73 if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) 1154.08/294.73 minsort(x) -> mins(x, nil, nil) 1154.08/294.73 mins(x, y, z) -> if(null(x), x, y, z) 1154.08/294.73 if(true, x, y, z) -> z 1154.08/294.73 if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) 1154.08/294.73 if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) 1154.08/294.73 if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) 1154.08/294.73 1154.08/294.73 S is empty. 1154.08/294.73 Rewrite Strategy: INNERMOST 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1154.08/294.73 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1154.08/294.73 1154.08/294.73 The rewrite sequence 1154.08/294.73 1154.08/294.73 le(s(x), s(y)) ->^+ le(x, y) 1154.08/294.73 1154.08/294.73 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1154.08/294.73 1154.08/294.73 The pumping substitution is [x / s(x), y / s(y)]. 1154.08/294.73 1154.08/294.73 The result substitution is [ ]. 1154.08/294.73 1154.08/294.73 1154.08/294.73 1154.08/294.73 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (4) 1154.08/294.73 Complex Obligation (BEST) 1154.08/294.73 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (5) 1154.08/294.73 Obligation: 1154.08/294.73 Proved the lower bound n^1 for the following obligation: 1154.08/294.73 1154.08/294.73 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1154.08/294.73 1154.08/294.73 1154.08/294.73 The TRS R consists of the following rules: 1154.08/294.73 1154.08/294.73 eq(0, 0) -> true 1154.08/294.73 eq(0, s(x)) -> false 1154.08/294.73 eq(s(x), 0) -> false 1154.08/294.73 eq(s(x), s(y)) -> eq(x, y) 1154.08/294.73 le(0, y) -> true 1154.08/294.73 le(s(x), 0) -> false 1154.08/294.73 le(s(x), s(y)) -> le(x, y) 1154.08/294.73 app(nil, y) -> y 1154.08/294.73 app(add(n, x), y) -> add(n, app(x, y)) 1154.08/294.73 min(add(n, nil)) -> n 1154.08/294.73 min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) 1154.08/294.73 if_min(true, add(n, add(m, x))) -> min(add(n, x)) 1154.08/294.73 if_min(false, add(n, add(m, x))) -> min(add(m, x)) 1154.08/294.73 head(add(n, x)) -> n 1154.08/294.73 tail(add(n, x)) -> x 1154.08/294.73 tail(nil) -> nil 1154.08/294.73 null(nil) -> true 1154.08/294.73 null(add(n, x)) -> false 1154.08/294.73 rm(n, nil) -> nil 1154.08/294.73 rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) 1154.08/294.73 if_rm(true, n, add(m, x)) -> rm(n, x) 1154.08/294.73 if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) 1154.08/294.73 minsort(x) -> mins(x, nil, nil) 1154.08/294.73 mins(x, y, z) -> if(null(x), x, y, z) 1154.08/294.73 if(true, x, y, z) -> z 1154.08/294.73 if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) 1154.08/294.73 if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) 1154.08/294.73 if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) 1154.08/294.73 1154.08/294.73 S is empty. 1154.08/294.73 Rewrite Strategy: INNERMOST 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (6) LowerBoundPropagationProof (FINISHED) 1154.08/294.73 Propagated lower bound. 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (7) 1154.08/294.73 BOUNDS(n^1, INF) 1154.08/294.73 1154.08/294.73 ---------------------------------------- 1154.08/294.73 1154.08/294.73 (8) 1154.08/294.73 Obligation: 1154.08/294.73 Analyzing the following TRS for decreasing loops: 1154.08/294.73 1154.08/294.73 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1154.08/294.73 1154.08/294.73 1154.08/294.73 The TRS R consists of the following rules: 1154.08/294.73 1154.08/294.73 eq(0, 0) -> true 1154.08/294.73 eq(0, s(x)) -> false 1154.08/294.73 eq(s(x), 0) -> false 1154.08/294.73 eq(s(x), s(y)) -> eq(x, y) 1154.08/294.73 le(0, y) -> true 1154.08/294.73 le(s(x), 0) -> false 1154.08/294.73 le(s(x), s(y)) -> le(x, y) 1154.08/294.73 app(nil, y) -> y 1154.08/294.73 app(add(n, x), y) -> add(n, app(x, y)) 1154.08/294.73 min(add(n, nil)) -> n 1154.08/294.73 min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) 1154.08/294.73 if_min(true, add(n, add(m, x))) -> min(add(n, x)) 1154.08/294.73 if_min(false, add(n, add(m, x))) -> min(add(m, x)) 1154.08/294.73 head(add(n, x)) -> n 1154.08/294.73 tail(add(n, x)) -> x 1154.08/294.73 tail(nil) -> nil 1154.08/294.73 null(nil) -> true 1154.08/294.73 null(add(n, x)) -> false 1154.08/294.73 rm(n, nil) -> nil 1154.08/294.73 rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) 1154.08/294.73 if_rm(true, n, add(m, x)) -> rm(n, x) 1154.08/294.73 if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) 1154.08/294.73 minsort(x) -> mins(x, nil, nil) 1154.08/294.73 mins(x, y, z) -> if(null(x), x, y, z) 1154.08/294.73 if(true, x, y, z) -> z 1154.08/294.73 if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) 1154.08/294.73 if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) 1154.08/294.73 if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) 1154.08/294.73 1154.08/294.73 S is empty. 1154.08/294.73 Rewrite Strategy: INNERMOST 1154.15/294.79 EOF