The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

subsets(Cons(x, xs)) → subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
subsets(Nil) → Cons(Nil, Nil)
mapconsapp(x', Cons(x, xs), rest) → Cons(Cons(x', x), mapconsapp(x', xs, rest))
mapconsapp(x, Nil, rest) → rest
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)

The (relative) TRS S consists of the following rules:

subsets[Ite][True][Let](Cons(x, xs), subs) → mapconsapp(x, subs, subs)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
subsets(Cons(x, xs)) →⁺ subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [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:

subsets(Cons(x, xs)) → subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
subsets(Nil) → Cons(Nil, Nil)
mapconsapp(x', Cons(x, xs), rest) → Cons(Cons(x', x), mapconsapp(x', xs, rest))
mapconsapp(x, Nil, rest) → rest
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)

The (relative) TRS S consists of the following rules:

subsets[Ite][True][Let](Cons(x, xs), subs) → mapconsapp(x, subs, subs)

Rewrite Strategy: INNERMOST

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0
mapconsapp/0

(6) Obligation:

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

subsets(Cons(xs)) → subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) → Cons(Nil)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) → rest
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)

The (relative) TRS S consists of the following rules:

subsets[Ite][True][Let](Cons(xs), subs) → mapconsapp(subs, subs)

Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Innermost TRS:
Rules:
subsets(Cons(xs)) → subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) → Cons(Nil)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) → rest
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) → mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
subsets, mapconsapp

(10) Obligation:

Innermost TRS:
Rules:
subsets(Cons(xs)) → subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) → Cons(Nil)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) → rest
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) → mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
subsets, mapconsapp

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

Proved the following rewrite lemma:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
subsets(gen_Cons:Nil3_0(+(1, 0)))

Induction Step:
subsets(gen_Cons:Nil3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
subsets[Ite][True][Let](Cons(gen_Cons:Nil3_0(+(1, n5_0))), subsets(gen_Cons:Nil3_0(+(1, n5_0)))) →_IH
subsets[Ite][True][Let](Cons(gen_Cons:Nil3_0(+(1, n5_0))), *4_0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
subsets(Cons(xs)) → subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) → Cons(Nil)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) → rest
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) → mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
mapconsapp

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mapconsapp(gen_Cons:Nil3_0(n3503_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n3503_0, b)), rt ∈ Ω(1 + n3503₀)

Induction Base:
mapconsapp(gen_Cons:Nil3_0(0), gen_Cons:Nil3_0(b)) →_R^Ω(1)
gen_Cons:Nil3_0(b)

Induction Step:
mapconsapp(gen_Cons:Nil3_0(+(n3503_0, 1)), gen_Cons:Nil3_0(b)) →_R^Ω(1)
Cons(mapconsapp(gen_Cons:Nil3_0(n3503_0), gen_Cons:Nil3_0(b))) →_IH
Cons(gen_Cons:Nil3_0(+(b, c3504_0)))

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

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
subsets(Cons(xs)) → subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) → Cons(Nil)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) → rest
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) → mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
mapconsapp(gen_Cons:Nil3_0(n3503_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n3503_0, b)), rt ∈ Ω(1 + n3503₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

Innermost TRS:
Rules:
subsets(Cons(xs)) → subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) → Cons(Nil)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) → rest
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) → mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
mapconsapp(gen_Cons:Nil3_0(n3503_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n3503_0, b)), rt ∈ Ω(1 + n3503₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
subsets(Cons(xs)) → subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) → Cons(Nil)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) → rest
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) → mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(24) BOUNDS(n^1, INF)