Runtime Complexity (full) proof of /tmp/tmpwBJB5D/ExAppendixB_AEL03

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 1579 ms)

↳2 BOUNDS(n^1, INF)

↳3 RenamingProof (⇔, 0 ms)

↳4 CpxRelTRS

    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 typed CpxTrs

    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)

    ↳8 typed CpxTrs

    ↳9 RewriteLemmaProof (LOWER BOUND(ID), 767 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

            ↳12 RewriteLemmaProof (LOWER BOUND(ID), 267 ms)

            ↳13 BEST

                ↳14 typed CpxTrs

                    ↳15 NoRewriteLemmaProof (LOWER BOUND(ID), 444 ms)

                    ↳16 typed CpxTrs

                    ↳17 NoRewriteLemmaProof (LOWER BOUND(ID), 270 ms)

                    ↳18 typed CpxTrs

                    ↳19 LowerBoundsProof (⇔, 0 ms)

                    ↳20 BOUNDS(n^2, INF)

                ↳21 typed CpxTrs

                    ↳22 LowerBoundsProof (⇔, 0 ms)

                    ↳23 BOUNDS(n^2, INF)

        ↳24 typed CpxTrs

            ↳25 LowerBoundsProof (⇔, 0 ms)

            ↳26 BOUNDS(n^1, INF)

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

from(X) → cons(X, n__from(s(X)))
2ndspos(0, Z) → rnil
2ndspos(s(N), cons(X, Z)) → 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) → rnil
2ndsneg(s(N), cons(X, Z)) → 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0))
plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0, Y) → 0
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
activate(n__from(X)) → from(X)
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
2ndspos(s(s(N1134_1)), cons(X, cons(Y400_1, cons2(X1135_1, cons(Y1136_1, Z1137_1))))) →⁺ rcons(posrecip(Y400_1), rcons(negrecip(Y1136_1), 2ndspos(N1134_1, Z1137_1)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [N1134_1 / s(s(N1134_1)), Z1137_1 / cons(X, cons(Y400_1, cons2(X1135_1, cons(Y1136_1, Z1137_1))))].
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)))
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(X, Z)) → 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(X, Z)) → 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0'))
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0', Y) → 0'
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
activate(n__from(X)) → from(X)
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)))
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(X, Z)) → 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(X, Z)) → 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0'))
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0', Y) → 0'
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X

Types:
from :: s:0' → n__from:cons:cons2
cons :: s:0' → n__from:cons:cons2 → n__from:cons:cons2
n__from :: s:0' → n__from:cons:cons2
s :: s:0' → s:0'
2ndspos :: s:0' → n__from:cons:cons2 → rnil:rcons
0' :: s:0'
rnil :: rnil:rcons
cons2 :: s:0' → n__from:cons:cons2 → n__from:cons:cons2
activate :: n__from:cons:cons2 → n__from:cons:cons2
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:0' → posrecip:negrecip
2ndsneg :: s:0' → n__from:cons:cons2 → rnil:rcons
negrecip :: s:0' → posrecip:negrecip
pi :: s:0' → rnil:rcons
plus :: s:0' → s:0' → s:0'
times :: s:0' → s:0' → s:0'
square :: s:0' → s:0'
hole_n__from:cons:cons21_0 :: n__from:cons:cons2
hole_s:0'2_0 :: s:0'
hole_rnil:rcons3_0 :: rnil:rcons
hole_posrecip:negrecip4_0 :: posrecip:negrecip
gen_n__from:cons:cons25_0 :: Nat → n__from:cons:cons2
gen_s:0'6_0 :: Nat → s:0'
gen_rnil:rcons7_0 :: Nat → rnil:rcons

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
2ndspos, 2ndsneg, plus, times

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg
plus < times

(8) Obligation:

Generator Equations:
gen_n__from:cons:cons25_0(0) ⇔ n__from(0')
gen_n__from:cons:cons25_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:cons25_0(x))
gen_s:0'6_0(0) ⇔ 0'
gen_s:0'6_0(+(x, 1)) ⇔ s(gen_s:0'6_0(x))
gen_rnil:rcons7_0(0) ⇔ rnil
gen_rnil:rcons7_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons7_0(x))

The following defined symbols remain to be analysed:
plus, 2ndspos, 2ndsneg, times

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg
plus < times

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
plus(gen_s:0'6_0(n9_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n9_0, b)), rt ∈ Ω(1 + n9₀)

Induction Base:
plus(gen_s:0'6_0(0), gen_s:0'6_0(b)) →_R^Ω(1)
gen_s:0'6_0(b)

Induction Step:
plus(gen_s:0'6_0(+(n9_0, 1)), gen_s:0'6_0(b)) →_R^Ω(1)
s(plus(gen_s:0'6_0(n9_0), gen_s:0'6_0(b))) →_IH
s(gen_s:0'6_0(+(b, c10_0)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

Lemmas:
plus(gen_s:0'6_0(n9_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n9_0, b)), rt ∈ Ω(1 + n9₀)

The following defined symbols remain to be analysed:
times, 2ndspos, 2ndsneg

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
times(gen_s:0'6_0(n928_0), gen_s:0'6_0(b)) → gen_s:0'6_0(*(n928_0, b)), rt ∈ Ω(1 + b·n928₀ + n928₀)

Induction Base:
times(gen_s:0'6_0(0), gen_s:0'6_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_s:0'6_0(+(n928_0, 1)), gen_s:0'6_0(b)) →_R^Ω(1)
plus(gen_s:0'6_0(b), times(gen_s:0'6_0(n928_0), gen_s:0'6_0(b))) →_IH
plus(gen_s:0'6_0(b), gen_s:0'6_0(*(c929_0, b))) →_L^{Ω(1 + b)}
gen_s:0'6_0(+(b, *(n928_0, b)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(13) Complex Obligation (BEST)

(14) Obligation:

Lemmas:
plus(gen_s:0'6_0(n9_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n9_0, b)), rt ∈ Ω(1 + n9₀)
times(gen_s:0'6_0(n928_0), gen_s:0'6_0(b)) → gen_s:0'6_0(*(n928_0, b)), rt ∈ Ω(1 + b·n928₀ + n928₀)

The following defined symbols remain to be analysed:
2ndsneg, 2ndspos

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol 2ndsneg.

(16) Obligation:

The following defined symbols remain to be analysed:
2ndspos

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol 2ndspos.

(18) Obligation:

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_s:0'6_0(n928_0), gen_s:0'6_0(b)) → gen_s:0'6_0(*(n928_0, b)), rt ∈ Ω(1 + b·n928₀ + n928₀)

(20) BOUNDS(n^2, INF)

(21) Obligation:

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_s:0'6_0(n928_0), gen_s:0'6_0(b)) → gen_s:0'6_0(*(n928_0, b)), rt ∈ Ω(1 + b·n928₀ + n928₀)

(23) BOUNDS(n^2, INF)

(24) Obligation:

Lemmas:
plus(gen_s:0'6_0(n9_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n9_0, b)), rt ∈ Ω(1 + n9₀)

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_s:0'6_0(n9_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n9_0, b)), rt ∈ Ω(1 + n9₀)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

(4) Obligation:

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

(6) Obligation:

(7) OrderProof (LOWER BOUND(ID) transformation)

(8) Obligation:

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Complex Obligation (BEST)

(11) Obligation:

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(13) Complex Obligation (BEST)

(14) Obligation:

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(16) Obligation:

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(18) Obligation:

(19) LowerBoundsProof (EQUIVALENT transformation)

(20) BOUNDS(n^2, INF)

(21) Obligation:

(22) LowerBoundsProof (EQUIVALENT transformation)

(23) BOUNDS(n^2, INF)

(24) Obligation:

(25) LowerBoundsProof (EQUIVALENT transformation)

(26) BOUNDS(n^1, INF)