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), 148 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), 176 ms)
↳18 CpxRNTS
↳19 IntTrsBoundProof (UPPER BOUND(ID), 24 ms)
↳20 CpxRNTS
↳21 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 577 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 223 ms)
↳26 CpxRNTS
↳27 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 203 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 63 ms)
↳32 CpxRNTS
↳33 FinalProof (⇔, 0 ms)
↳34 BOUNDS(1, n^1)
from(X) → cons(X, n__from(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
from(X) → cons(X, n__from(s(X))) [1]
sel(0, cons(X, Y)) → X [1]
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) [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]
sel(0, cons(X, Y)) → X [1]
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) [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 sel :: s:0 → n__from:cons → s:0 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:
sel
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
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(X, from(X')) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0
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'
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
sel(z, z') -{ 3 }→ sel(X, 1 + X'') :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(X, 1 + X'' + (1 + (1 + X''))) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = 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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
{ activate } { sel } { from } |
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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = 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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] sel: 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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] sel: runtime: O(n1) [1 + 8·z], 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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ -5 + 8·z }→ s :|: s >= 0, s <= inf, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ -4 + 8·z }→ s' :|: s' >= 0, s' <= inf', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ -4 + 8·z }→ s'' :|: s'' >= 0, s'' <= inf'', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] sel: runtime: O(n1) [1 + 8·z], 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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ -5 + 8·z }→ s :|: s >= 0, s <= inf, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ -4 + 8·z }→ s' :|: s' >= 0, s' <= inf', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ -4 + 8·z }→ s'' :|: s'' >= 0, s'' <= inf'', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] sel: runtime: O(n1) [1 + 8·z], size: EXP 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'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ -5 + 8·z }→ s :|: s >= 0, s <= inf, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ -4 + 8·z }→ s' :|: s' >= 0, s' <= inf', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ -4 + 8·z }→ s'' :|: s'' >= 0, s'' <= inf'', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] sel: runtime: O(n1) [1 + 8·z], size: EXP from: runtime: O(1) [1], size: O(n1) [3 + 2·z] |