Runtime Complexity (full) proof of /tmp/tmpBu4zrT/Ex15_Luc98

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

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 391 ms)

↳2 BOUNDS(2^n, 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 NoRewriteLemmaProof (LOWER BOUND(ID), 168 ms)

    ↳10 typed CpxTrs

    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 262 ms)

    ↳12 BEST

        ↳13 typed CpxTrs

            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

            ↳15 typed CpxTrs

            ↳16 LowerBoundsProof (⇔, 0 ms)

            ↳17 BOUNDS(n^1, INF)

        ↳18 typed CpxTrs

            ↳19 LowerBoundsProof (⇔, 0 ms)

            ↳20 BOUNDS(n^1, INF)

(0) Obligation:

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

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
from(n__from(X7207_3)) →⁺ cons(from(X7207_3), n__from(n__s(from(X7207_3))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X7207_3 / n__from(X7207_3)].
The result substitution is [ ].

The rewrite sequence
from(n__from(X7207_3)) →⁺ cons(from(X7207_3), n__from(n__s(from(X7207_3))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X7207_3 / n__from(X7207_3)].
The result substitution is [ ].

(2) BOUNDS(2^n, 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:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0', X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0', X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

Types:
and :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
true :: true:false:0':n__add:nil:cons:n__first:n__s:n__from
activate :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
false :: true:false:0':n__add:nil:cons:n__first:n__s:n__from
if :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
add :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
0' :: true:false:0':n__add:nil:cons:n__first:n__s:n__from
s :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
n__add :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
first :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
nil :: true:false:0':n__add:nil:cons:n__first:n__s:n__from
cons :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
n__first :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
from :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
n__from :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
n__s :: true:false:0':n__add:nil:cons:n__first:n__s:n__from → true:false:0':n__add:nil:cons:n__first:n__s:n__from
hole_true:false:0':n__add:nil:cons:n__first:n__s:n__from1_0 :: true:false:0':n__add:nil:cons:n__first:n__s:n__from
gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0 :: Nat → true:false:0':n__add:nil:cons:n__first:n__s:n__from

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

Heuristically decided to analyse the following defined symbols:
activate, from

They will be analysed ascendingly in the following order:
activate = from

(8) Obligation:

Generator Equations:
gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(0) ⇔ true
gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(x, 1)) ⇔ n__add(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(x), true)

The following defined symbols remain to be analysed:
from, activate

They will be analysed ascendingly in the following order:
activate = from

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol from.

(10) Obligation:

The following defined symbols remain to be analysed:
activate

They will be analysed ascendingly in the following order:
activate = from

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, n2094_0))) → *3_0, rt ∈ Ω(n2094₀)

Induction Base:
activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, 0)))

Induction Step:
activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, +(n2094_0, 1)))) →_R^Ω(1)
add(activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, n2094_0))), true) →_IH
add(*3_0, true)

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

(12) Complex Obligation (BEST)

(13) Obligation:

Lemmas:
activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, n2094_0))) → *3_0, rt ∈ Ω(n2094₀)

The following defined symbols remain to be analysed:
from

They will be analysed ascendingly in the following order:
activate = from

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol from.

(15) Obligation:

Lemmas:
activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, n2094_0))) → *3_0, rt ∈ Ω(n2094₀)

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, n2094_0))) → *3_0, rt ∈ Ω(n2094₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Lemmas:
activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, n2094_0))) → *3_0, rt ∈ Ω(n2094₀)

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_true:false:0':n__add:nil:cons:n__first:n__s:n__from2_0(+(1, n2094_0))) → *3_0, rt ∈ Ω(n2094₀)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(2^n, 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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Obligation:

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(12) Complex Obligation (BEST)

(13) Obligation:

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(15) Obligation:

(16) LowerBoundsProof (EQUIVALENT transformation)

(17) BOUNDS(n^1, INF)

(18) Obligation:

(19) LowerBoundsProof (EQUIVALENT transformation)

(20) BOUNDS(n^1, INF)