0 CpxTRS
↳1 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxWeightedTrs
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxTypedWeightedTrs
↳5 CompletionProof (UPPER BOUND(ID), 0 ms)
↳6 CpxTypedWeightedCompleteTrs
↳7 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)
↳10 CpxRNTS
↳11 InliningProof (UPPER BOUND(ID), 123 ms)
↳12 CpxRNTS
↳13 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CpxRNTS
↳15 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CpxRNTS
↳17 IntTrsBoundProof (UPPER BOUND(ID), 166 ms)
↳18 CpxRNTS
↳19 IntTrsBoundProof (UPPER BOUND(ID), 32 ms)
↳20 CpxRNTS
↳21 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 234 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 43 ms)
↳26 CpxRNTS
↳27 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 501 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 202 ms)
↳32 CpxRNTS
↳33 FinalProof (⇔, 0 ms)
↳34 BOUNDS(1, n^1)
from(X) → cons(X, n__from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
from(X) → cons(X, n__from(s(X))) [1]
after(0, XS) → XS [1]
after(s(N), cons(X, XS)) → after(N, activate(XS)) [1]
from(X) → n__from(X) [1]
activate(n__from(X)) → from(X) [1]
activate(X) → X [1]
from(X) → cons(X, n__from(s(X))) [1]
after(0, XS) → XS [1]
after(s(N), cons(X, XS)) → after(N, activate(XS)) [1]
from(X) → n__from(X) [1]
activate(n__from(X)) → from(X) [1]
activate(X) → X [1]
from :: s:0 → n__from:cons cons :: s:0 → n__from:cons → n__from:cons n__from :: s:0 → n__from:cons s :: s:0 → s:0 after :: s:0 → n__from:cons → n__from:cons 0 :: s:0 activate :: n__from:cons → n__from:cons |
(a) The obligation is a constructor system where every type has a constant constructor,
(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
after
activate
from
const
Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules:
The TRS has the following type information:
Rewrite Strategy: INNERMOST |
Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules:
The TRS has the following type information:
Rewrite Strategy: INNERMOST |
0 => 0
const => 0
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ from(X) :|: z = 1 + X, X >= 0
after(z, z') -{ 1 }→ XS :|: z' = XS, z = 0, XS >= 0
after(z, z') -{ 2 }→ after(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
after(z, z') -{ 2 }→ after(N, from(X')) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
after(z, z') -{ 1 }→ XS :|: z' = XS, z = 0, XS >= 0
after(z, z') -{ 2 }→ after(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
after(z, z') -{ 3 }→ after(N, 1 + X'') :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(N, 1 + X'' + (1 + (1 + X''))) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
{ activate } { from } { after } |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate: runtime: ?, size: O(n1) [1 + 2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] from: runtime: ?, size: O(n1) [3 + 2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] from: runtime: O(1) [1], size: O(n1) [3 + 2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] from: runtime: O(1) [1], size: O(n1) [3 + 2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] from: runtime: O(1) [1], size: O(n1) [3 + 2·z] after: runtime: ?, size: EXP |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] from: runtime: O(1) [1], size: O(n1) [3 + 2·z] after: runtime: O(n1) [1 + 8·z], size: EXP |