(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(s(X)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0) → 0
minus(s(X), s(Y)) → minus(X, Y)
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
sel(s(N), cons(X, XS)) →+ sel(N, XS)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [N / s(N), XS / cons(X, XS)].
The result substitution is [ ].
(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(s(X)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X
Types:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
sel,
activate,
minus,
quotThey will be analysed ascendingly in the following order:
activate < sel
quot < activate
minus < quot
(8) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
n__from(
s(
X)))
sel(
0',
cons(
X,
XS)) →
Xsel(
s(
N),
cons(
X,
XS)) →
sel(
N,
activate(
XS))
minus(
X,
0') →
0'minus(
s(
X),
s(
Y)) →
minus(
X,
Y)
quot(
0',
s(
Y)) →
0'quot(
s(
X),
s(
Y)) →
s(
quot(
minus(
X,
Y),
s(
Y)))
zWquot(
XS,
nil) →
nilzWquot(
nil,
XS) →
nilzWquot(
cons(
X,
XS),
cons(
Y,
YS)) →
cons(
quot(
X,
Y),
n__zWquot(
activate(
XS),
activate(
YS)))
from(
X) →
n__from(
X)
zWquot(
X1,
X2) →
n__zWquot(
X1,
X2)
activate(
n__from(
X)) →
from(
X)
activate(
n__zWquot(
X1,
X2)) →
zWquot(
X1,
X2)
activate(
X) →
XTypes:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'
Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
The following defined symbols remain to be analysed:
minus, sel, activate, quot
They will be analysed ascendingly in the following order:
activate < sel
quot < activate
minus < quot
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
minus(
gen_s:0'4_0(
n6_0),
gen_s:0'4_0(
n6_0)) →
gen_s:0'4_0(
0), rt ∈ Ω(1 + n6
0)
Induction Base:
minus(gen_s:0'4_0(0), gen_s:0'4_0(0)) →RΩ(1)
0'
Induction Step:
minus(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) →RΩ(1)
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) →IH
gen_s:0'4_0(0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
n__from(
s(
X)))
sel(
0',
cons(
X,
XS)) →
Xsel(
s(
N),
cons(
X,
XS)) →
sel(
N,
activate(
XS))
minus(
X,
0') →
0'minus(
s(
X),
s(
Y)) →
minus(
X,
Y)
quot(
0',
s(
Y)) →
0'quot(
s(
X),
s(
Y)) →
s(
quot(
minus(
X,
Y),
s(
Y)))
zWquot(
XS,
nil) →
nilzWquot(
nil,
XS) →
nilzWquot(
cons(
X,
XS),
cons(
Y,
YS)) →
cons(
quot(
X,
Y),
n__zWquot(
activate(
XS),
activate(
YS)))
from(
X) →
n__from(
X)
zWquot(
X1,
X2) →
n__zWquot(
X1,
X2)
activate(
n__from(
X)) →
from(
X)
activate(
n__zWquot(
X1,
X2)) →
zWquot(
X1,
X2)
activate(
X) →
XTypes:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'
Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)
Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
The following defined symbols remain to be analysed:
quot, sel, activate
They will be analysed ascendingly in the following order:
activate < sel
quot < activate
(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol quot.
(13) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
n__from(
s(
X)))
sel(
0',
cons(
X,
XS)) →
Xsel(
s(
N),
cons(
X,
XS)) →
sel(
N,
activate(
XS))
minus(
X,
0') →
0'minus(
s(
X),
s(
Y)) →
minus(
X,
Y)
quot(
0',
s(
Y)) →
0'quot(
s(
X),
s(
Y)) →
s(
quot(
minus(
X,
Y),
s(
Y)))
zWquot(
XS,
nil) →
nilzWquot(
nil,
XS) →
nilzWquot(
cons(
X,
XS),
cons(
Y,
YS)) →
cons(
quot(
X,
Y),
n__zWquot(
activate(
XS),
activate(
YS)))
from(
X) →
n__from(
X)
zWquot(
X1,
X2) →
n__zWquot(
X1,
X2)
activate(
n__from(
X)) →
from(
X)
activate(
n__zWquot(
X1,
X2)) →
zWquot(
X1,
X2)
activate(
X) →
XTypes:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'
Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)
Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
The following defined symbols remain to be analysed:
activate, sel
They will be analysed ascendingly in the following order:
activate < sel
(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(15) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
n__from(
s(
X)))
sel(
0',
cons(
X,
XS)) →
Xsel(
s(
N),
cons(
X,
XS)) →
sel(
N,
activate(
XS))
minus(
X,
0') →
0'minus(
s(
X),
s(
Y)) →
minus(
X,
Y)
quot(
0',
s(
Y)) →
0'quot(
s(
X),
s(
Y)) →
s(
quot(
minus(
X,
Y),
s(
Y)))
zWquot(
XS,
nil) →
nilzWquot(
nil,
XS) →
nilzWquot(
cons(
X,
XS),
cons(
Y,
YS)) →
cons(
quot(
X,
Y),
n__zWquot(
activate(
XS),
activate(
YS)))
from(
X) →
n__from(
X)
zWquot(
X1,
X2) →
n__zWquot(
X1,
X2)
activate(
n__from(
X)) →
from(
X)
activate(
n__zWquot(
X1,
X2)) →
zWquot(
X1,
X2)
activate(
X) →
XTypes:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'
Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)
Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
The following defined symbols remain to be analysed:
sel
(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol sel.
(17) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
n__from(
s(
X)))
sel(
0',
cons(
X,
XS)) →
Xsel(
s(
N),
cons(
X,
XS)) →
sel(
N,
activate(
XS))
minus(
X,
0') →
0'minus(
s(
X),
s(
Y)) →
minus(
X,
Y)
quot(
0',
s(
Y)) →
0'quot(
s(
X),
s(
Y)) →
s(
quot(
minus(
X,
Y),
s(
Y)))
zWquot(
XS,
nil) →
nilzWquot(
nil,
XS) →
nilzWquot(
cons(
X,
XS),
cons(
Y,
YS)) →
cons(
quot(
X,
Y),
n__zWquot(
activate(
XS),
activate(
YS)))
from(
X) →
n__from(
X)
zWquot(
X1,
X2) →
n__zWquot(
X1,
X2)
activate(
n__from(
X)) →
from(
X)
activate(
n__zWquot(
X1,
X2)) →
zWquot(
X1,
X2)
activate(
X) →
XTypes:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'
Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)
Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
No more defined symbols left to analyse.
(18) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)
(19) BOUNDS(n^1, INF)
(20) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
n__from(
s(
X)))
sel(
0',
cons(
X,
XS)) →
Xsel(
s(
N),
cons(
X,
XS)) →
sel(
N,
activate(
XS))
minus(
X,
0') →
0'minus(
s(
X),
s(
Y)) →
minus(
X,
Y)
quot(
0',
s(
Y)) →
0'quot(
s(
X),
s(
Y)) →
s(
quot(
minus(
X,
Y),
s(
Y)))
zWquot(
XS,
nil) →
nilzWquot(
nil,
XS) →
nilzWquot(
cons(
X,
XS),
cons(
Y,
YS)) →
cons(
quot(
X,
Y),
n__zWquot(
activate(
XS),
activate(
YS)))
from(
X) →
n__from(
X)
zWquot(
X1,
X2) →
n__zWquot(
X1,
X2)
activate(
n__from(
X)) →
from(
X)
activate(
n__zWquot(
X1,
X2)) →
zWquot(
X1,
X2)
activate(
X) →
XTypes:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'
Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)
Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
No more defined symbols left to analyse.
(21) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)
(22) BOUNDS(n^1, INF)