KILLEDRuntime Complexity (full) proof of /tmp/tmp2VRKmH/Ex3_3_25_Bor03.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 RenamingProof (⇔, 0 ms)↳2 CpxRelTRS↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳4 typed CpxTrs↳5 OrderProof (LOWER BOUND(ID), 0 ms)↳6 typed CpxTrs↳7 RewriteLemmaProof (LOWER BOUND(ID), 854 ms)↳8 BEST↳9 typed CpxTrs↳10 NoRewriteLemmaProof (LOWER BOUND(ID), 178 ms)↳11 typed CpxTrs↳12 typed CpxTrs(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Rewrite Strategy: FULL(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
S is empty.
Rewrite Strategy: FULL(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(4) Obligation:
TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Types:
app :: nil:cons:s → nil:cons:s → nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s → nil:cons:s → nil:cons:s
from :: nil:cons:s → nil:cons:s
s :: nil:cons:s → nil:cons:s
zWadr :: nil:cons:s → nil:cons:s → nil:cons:s
prefix :: nil:cons:s → nil:cons:s
hole_nil:cons:s1_0 :: nil:cons:s
gen_nil:cons:s2_0 :: Nat → nil:cons:s(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
app, from, zWadr, prefixThey will be analysed ascendingly in the following order:
app < zWadr
zWadr < prefix(6) Obligation:
TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Types:
app :: nil:cons:s → nil:cons:s → nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s → nil:cons:s → nil:cons:s
from :: nil:cons:s → nil:cons:s
s :: nil:cons:s → nil:cons:s
zWadr :: nil:cons:s → nil:cons:s → nil:cons:s
prefix :: nil:cons:s → nil:cons:s
hole_nil:cons:s1_0 :: nil:cons:s
gen_nil:cons:s2_0 :: Nat → nil:cons:sGenerator Equations:
gen_nil:cons:s2_0(0) ⇔ nil
gen_nil:cons:s2_0(+(x, 1)) ⇔ cons(nil, gen_nil:cons:s2_0(x))The following defined symbols remain to be analysed:
app, from, zWadr, prefixThey will be analysed ascendingly in the following order:
app < zWadr
zWadr < prefix(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
app(gen_nil:cons:s2_0(n4_0), gen_nil:cons:s2_0(b)) → gen_nil:cons:s2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)Induction Base:
app(gen_nil:cons:s2_0(0), gen_nil:cons:s2_0(b)) →RΩ(1)
gen_nil:cons:s2_0(b)Induction Step:
app(gen_nil:cons:s2_0(+(n4_0, 1)), gen_nil:cons:s2_0(b)) →RΩ(1)
cons(nil, app(gen_nil:cons:s2_0(n4_0), gen_nil:cons:s2_0(b))) →IH
cons(nil, gen_nil:cons:s2_0(+(b, c5_0)))We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(8) Complex Obligation (BEST)
(9) Obligation:
TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Types:
app :: nil:cons:s → nil:cons:s → nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s → nil:cons:s → nil:cons:s
from :: nil:cons:s → nil:cons:s
s :: nil:cons:s → nil:cons:s
zWadr :: nil:cons:s → nil:cons:s → nil:cons:s
prefix :: nil:cons:s → nil:cons:s
hole_nil:cons:s1_0 :: nil:cons:s
gen_nil:cons:s2_0 :: Nat → nil:cons:sLemmas:
app(gen_nil:cons:s2_0(n4_0), gen_nil:cons:s2_0(b)) → gen_nil:cons:s2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)Generator Equations:
gen_nil:cons:s2_0(0) ⇔ nil
gen_nil:cons:s2_0(+(x, 1)) ⇔ cons(nil, gen_nil:cons:s2_0(x))The following defined symbols remain to be analysed:
from, zWadr, prefixThey will be analysed ascendingly in the following order:
zWadr < prefix(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol from.(11) Obligation:
TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Types:
app :: nil:cons:s → nil:cons:s → nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s → nil:cons:s → nil:cons:s
from :: nil:cons:s → nil:cons:s
s :: nil:cons:s → nil:cons:s
zWadr :: nil:cons:s → nil:cons:s → nil:cons:s
prefix :: nil:cons:s → nil:cons:s
hole_nil:cons:s1_0 :: nil:cons:s
gen_nil:cons:s2_0 :: Nat → nil:cons:sLemmas:
app(gen_nil:cons:s2_0(n4_0), gen_nil:cons:s2_0(b)) → gen_nil:cons:s2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)Generator Equations:
gen_nil:cons:s2_0(0) ⇔ nil
gen_nil:cons:s2_0(+(x, 1)) ⇔ cons(nil, gen_nil:cons:s2_0(x))The following defined symbols remain to be analysed:
zWadr, prefixThey will be analysed ascendingly in the following order:
zWadr < prefix(12) Obligation:
TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Types:
app :: nil:cons:s → nil:cons:s → nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s → nil:cons:s → nil:cons:s
from :: nil:cons:s → nil:cons:s
s :: nil:cons:s → nil:cons:s
zWadr :: nil:cons:s → nil:cons:s → nil:cons:s
prefix :: nil:cons:s → nil:cons:s
hole_nil:cons:s1_0 :: nil:cons:s
gen_nil:cons:s2_0 :: Nat → nil:cons:sLemmas:
app(gen_nil:cons:s2_0(n4_0), gen_nil:cons:s2_0(b)) → gen_nil:cons:s2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)Generator Equations:
gen_nil:cons:s2_0(0) ⇔ nil
gen_nil:cons:s2_0(+(x, 1)) ⇔ cons(nil, gen_nil:cons:s2_0(x))No more defined symbols left to analyse.