0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
↳3 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxWeightedTrs
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CpxTypedWeightedTrs
↳7 CompletionProof (UPPER BOUND(ID), 1 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CpxTypedWeightedCompleteTrs
↳11 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 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), 288 ms)
↳18 CpxRNTS
↳19 IntTrsBoundProof (UPPER BOUND(ID), 112 ms)
↳20 CpxRNTS
↳21 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 290 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 96 ms)
↳26 CpxRNTS
↳27 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 198 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 100 ms)
↳32 CpxRNTS
↳33 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳34 CpxRNTS
↳35 IntTrsBoundProof (UPPER BOUND(ID), 574 ms)
↳36 CpxRNTS
↳37 IntTrsBoundProof (UPPER BOUND(ID), 58 ms)
↳38 CpxRNTS
↳39 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳40 CpxRNTS
↳41 IntTrsBoundProof (UPPER BOUND(ID), 847 ms)
↳42 CpxRNTS
↳43 IntTrsBoundProof (UPPER BOUND(ID), 26 ms)
↳44 CpxRNTS
↳45 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳46 CpxRNTS
↳47 IntTrsBoundProof (UPPER BOUND(ID), 2481 ms)
↳48 CpxRNTS
↳49 IntTrsBoundProof (UPPER BOUND(ID), 124 ms)
↳50 CpxRNTS
↳51 FinalProof (⇔, 5 ms)
↳52 BOUNDS(1, n^3)
terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
half(0) → 0
half(s(0)) → 0
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X
half(s(0)) → 0
dbl(0) → 0
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
half(0) → 0
terms(X) → n__terms(X)
terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
first(0, X) → nil
dbl(s(X)) → s(s(dbl(X)))
activate(X) → X
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__terms(X)) → terms(X)
sqr(0) → 0
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(X1, X2) → n__first(X1, X2)
sqr(s(X)) → s(add(sqr(X), dbl(X)))
half(s(s(X))) → s(half(X))
half(s(0)) → 0 [1]
dbl(0) → 0 [1]
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1]
half(0) → 0 [1]
terms(X) → n__terms(X) [1]
terms(N) → cons(recip(sqr(N)), n__terms(s(N))) [1]
first(0, X) → nil [1]
dbl(s(X)) → s(s(dbl(X))) [1]
activate(X) → X [1]
activate(n__first(X1, X2)) → first(X1, X2) [1]
activate(n__terms(X)) → terms(X) [1]
sqr(0) → 0 [1]
add(0, X) → X [1]
add(s(X), Y) → s(add(X, Y)) [1]
first(X1, X2) → n__first(X1, X2) [1]
sqr(s(X)) → s(add(sqr(X), dbl(X))) [1]
half(s(s(X))) → s(half(X)) [1]
half(s(0)) → 0 [1]
dbl(0) → 0 [1]
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1]
half(0) → 0 [1]
terms(X) → n__terms(X) [1]
terms(N) → cons(recip(sqr(N)), n__terms(s(N))) [1]
first(0, X) → nil [1]
dbl(s(X)) → s(s(dbl(X))) [1]
activate(X) → X [1]
activate(n__first(X1, X2)) → first(X1, X2) [1]
activate(n__terms(X)) → terms(X) [1]
sqr(0) → 0 [1]
add(0, X) → X [1]
add(s(X), Y) → s(add(X, Y)) [1]
first(X1, X2) → n__first(X1, X2) [1]
sqr(s(X)) → s(add(sqr(X), dbl(X))) [1]
half(s(s(X))) → s(half(X)) [1]
half :: 0:s → 0:s s :: 0:s → 0:s 0 :: 0:s dbl :: 0:s → 0:s first :: 0:s → cons:n__first:n__terms:nil → cons:n__first:n__terms:nil cons :: recip → cons:n__first:n__terms:nil → cons:n__first:n__terms:nil n__first :: 0:s → cons:n__first:n__terms:nil → cons:n__first:n__terms:nil activate :: cons:n__first:n__terms:nil → cons:n__first:n__terms:nil terms :: 0:s → cons:n__first:n__terms:nil n__terms :: 0:s → cons:n__first:n__terms:nil recip :: 0:s → recip sqr :: 0:s → 0:s nil :: cons:n__first:n__terms:nil add :: 0:s → 0:s → 0:s |
(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:
half
first
terms
activate
sqr
dbl
add
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
nil => 0
const => 0
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ terms(X) :|: z = 1 + X, X >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ X :|: z' = X, X >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 }→ 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0
first(z, z') -{ 1 }→ 0 :|: z' = X, X >= 0, z = 0
first(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
first(z, z') -{ 1 }→ 1 + Y + (1 + X + activate(Z)) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(X) :|: z = 1 + (1 + X), X >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + add(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + add(1 + add(sqr(X'), dbl(X')), 1 + (1 + dbl(X'))) :|: X' >= 0, z = 1 + (1 + X')
terms(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
terms(z) -{ 1 }→ 1 + (1 + sqr(N)) + (1 + (1 + N)) :|: z = N, N >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 }→ 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + add(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
{ dbl } { add } { half } { sqr } { terms } { first, activate } |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 }→ 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + add(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 }→ 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + add(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: ?, size: O(n1) [2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 }→ 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + add(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + add(0, 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + add(0, 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: ?, size: O(n1) [z + z'] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + add(0, 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: ?, size: O(n1) [z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] sqr: runtime: ?, size: O(n2) [1 + 4·z + 4·z2] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 1 }→ 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] sqr: runtime: O(n3) [5 + 22·z + 8·z3], size: O(n2) [1 + 4·z + 4·z2] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ -99 + s5 + s6 + 120·z + -48·z2 + 8·z3 }→ 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= 1 * s5 + 1 * s', s7 >= 0, s7 <= 1 * (1 + s6) + 1 * (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 6 + 22·z + 8·z3 }→ 1 + (1 + s4) + (1 + (1 + z)) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] sqr: runtime: O(n3) [5 + 22·z + 8·z3], size: O(n2) [1 + 4·z + 4·z2] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ -99 + s5 + s6 + 120·z + -48·z2 + 8·z3 }→ 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= 1 * s5 + 1 * s', s7 >= 0, s7 <= 1 * (1 + s6) + 1 * (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 6 + 22·z + 8·z3 }→ 1 + (1 + s4) + (1 + (1 + z)) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] sqr: runtime: O(n3) [5 + 22·z + 8·z3], size: O(n2) [1 + 4·z + 4·z2] terms: runtime: ?, size: O(n2) [6 + 6·z + 4·z2] |
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ terms(z - 1) :|: z - 1 >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ -99 + s5 + s6 + 120·z + -48·z2 + 8·z3 }→ 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= 1 * s5 + 1 * s', s7 >= 0, s7 <= 1 * (1 + s6) + 1 * (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 6 + 22·z + 8·z3 }→ 1 + (1 + s4) + (1 + (1 + z)) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] sqr: runtime: O(n3) [5 + 22·z + 8·z3], size: O(n2) [1 + 4·z + 4·z2] terms: runtime: O(n3) [7 + 22·z + 8·z3], size: O(n2) [6 + 6·z + 4·z2] |
activate(z) -{ -22 + 46·z + -24·z2 + 8·z3 }→ s8 :|: s8 >= 0, s8 <= 6 * (z - 1) + 4 * ((z - 1) * (z - 1)) + 6, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ -99 + s5 + s6 + 120·z + -48·z2 + 8·z3 }→ 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= 1 * s5 + 1 * s', s7 >= 0, s7 <= 1 * (1 + s6) + 1 * (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 6 + 22·z + 8·z3 }→ 1 + (1 + s4) + (1 + (1 + z)) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] sqr: runtime: O(n3) [5 + 22·z + 8·z3], size: O(n2) [1 + 4·z + 4·z2] terms: runtime: O(n3) [7 + 22·z + 8·z3], size: O(n2) [6 + 6·z + 4·z2] |
activate(z) -{ -22 + 46·z + -24·z2 + 8·z3 }→ s8 :|: s8 >= 0, s8 <= 6 * (z - 1) + 4 * ((z - 1) * (z - 1)) + 6, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ -99 + s5 + s6 + 120·z + -48·z2 + 8·z3 }→ 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= 1 * s5 + 1 * s', s7 >= 0, s7 <= 1 * (1 + s6) + 1 * (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 6 + 22·z + 8·z3 }→ 1 + (1 + s4) + (1 + (1 + z)) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] sqr: runtime: O(n3) [5 + 22·z + 8·z3], size: O(n2) [1 + 4·z + 4·z2] terms: runtime: O(n3) [7 + 22·z + 8·z3], size: O(n2) [6 + 6·z + 4·z2] first: runtime: ?, size: INF activate: runtime: ?, size: INF |
activate(z) -{ -22 + 46·z + -24·z2 + 8·z3 }→ s8 :|: s8 >= 0, s8 <= 6 * (z - 1) + 4 * ((z - 1) * (z - 1)) + 6, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
dbl(z) -{ 1 }→ 0 :|: z = 0
dbl(z) -{ 1 + z }→ 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ -99 + s5 + s6 + 120·z + -48·z2 + 8·z3 }→ 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= 1 * s5 + 1 * s', s7 >= 0, s7 <= 1 * (1 + s6) + 1 * (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
terms(z) -{ 1 }→ 1 + z :|: z >= 0
terms(z) -{ 6 + 22·z + 8·z3 }→ 1 + (1 + s4) + (1 + (1 + z)) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0
dbl: runtime: O(n1) [1 + z], size: O(n1) [2·z] add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] half: runtime: O(n1) [1 + z], size: O(n1) [z] sqr: runtime: O(n3) [5 + 22·z + 8·z3], size: O(n2) [1 + 4·z + 4·z2] terms: runtime: O(n3) [7 + 22·z + 8·z3], size: O(n2) [6 + 6·z + 4·z2] first: runtime: O(n3) [3 + 48·z' + 8·z'3], size: INF activate: runtime: O(n3) [5 + 94·z + 16·z3], size: INF |