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

(0) Obligation:

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

rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Rewrite Strategy: INNERMOST

(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:

rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
rec, sent, no, top, check

They will be analysed ascendingly in the following order:
sent < rec
rec < top
rec < check
sent < check
no < check
check < top

(6) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

The following defined symbols remain to be analysed:
sent, rec, no, top, check

They will be analysed ascendingly in the following order:
sent < rec
rec < top
rec < check
sent < check
no < check
check < top

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
sent(gen_bot:up3_0(+(1, 0)))

Induction Step:
sent(gen_bot:up3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
up(sent(gen_bot:up3_0(+(1, n5_0)))) →_IH
up(*4_0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

The following defined symbols remain to be analysed:
rec, no, top, check

They will be analysed ascendingly in the following order:
rec < top
rec < check
no < check
check < top

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
rec(gen_bot:up3_0(+(1, n203_0))) → *4_0, rt ∈ Ω(n203₀)

Induction Base:
rec(gen_bot:up3_0(+(1, 0)))

Induction Step:
rec(gen_bot:up3_0(+(1, +(n203_0, 1)))) →_R^Ω(1)
up(rec(gen_bot:up3_0(+(1, n203_0)))) →_IH
up(*4_0)

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
rec(gen_bot:up3_0(+(1, n203_0))) → *4_0, rt ∈ Ω(n203₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

The following defined symbols remain to be analysed:
no, top, check

They will be analysed ascendingly in the following order:
no < check
check < top

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
no(gen_bot:up3_0(+(1, n160409_0))) → *4_0, rt ∈ Ω(n160409₀)

Induction Base:
no(gen_bot:up3_0(+(1, 0)))

Induction Step:
no(gen_bot:up3_0(+(1, +(n160409_0, 1)))) →_R^Ω(1)
up(no(gen_bot:up3_0(+(1, n160409_0)))) →_IH
up(*4_0)

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
rec(gen_bot:up3_0(+(1, n203_0))) → *4_0, rt ∈ Ω(n203₀)
no(gen_bot:up3_0(+(1, n160409_0))) → *4_0, rt ∈ Ω(n160409₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

The following defined symbols remain to be analysed:
check, top

They will be analysed ascendingly in the following order:
check < top

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
check(gen_bot:up3_0(+(1, n160809_0))) → *4_0, rt ∈ Ω(n160809₀)

Induction Base:
check(gen_bot:up3_0(+(1, 0)))

Induction Step:
check(gen_bot:up3_0(+(1, +(n160809_0, 1)))) →_R^Ω(1)
up(check(gen_bot:up3_0(+(1, n160809_0)))) →_IH
up(*4_0)

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

(17) Complex Obligation (BEST)

(18) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
rec(gen_bot:up3_0(+(1, n203_0))) → *4_0, rt ∈ Ω(n203₀)
no(gen_bot:up3_0(+(1, n160409_0))) → *4_0, rt ∈ Ω(n160409₀)
check(gen_bot:up3_0(+(1, n160809_0))) → *4_0, rt ∈ Ω(n160809₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

The following defined symbols remain to be analysed:
top

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
rec(gen_bot:up3_0(+(1, n203_0))) → *4_0, rt ∈ Ω(n203₀)
no(gen_bot:up3_0(+(1, n160409_0))) → *4_0, rt ∈ Ω(n160409₀)
check(gen_bot:up3_0(+(1, n160809_0))) → *4_0, rt ∈ Ω(n160809₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
rec(gen_bot:up3_0(+(1, n203_0))) → *4_0, rt ∈ Ω(n203₀)
no(gen_bot:up3_0(+(1, n160409_0))) → *4_0, rt ∈ Ω(n160409₀)
check(gen_bot:up3_0(+(1, n160809_0))) → *4_0, rt ∈ Ω(n160809₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
rec(gen_bot:up3_0(+(1, n203_0))) → *4_0, rt ∈ Ω(n203₀)
no(gen_bot:up3_0(+(1, n160409_0))) → *4_0, rt ∈ Ω(n160409₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(28) BOUNDS(n^1, INF)

(29) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
rec(gen_bot:up3_0(+(1, n203_0))) → *4_0, rt ∈ Ω(n203₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

Innermost TRS:
Rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

Types:
rec :: bot:up → bot:up
sent :: bot:up → bot:up
no :: bot:up → bot:up
bot :: bot:up
up :: bot:up → bot:up
top :: bot:up → top
check :: bot:up → bot:up
hole_bot:up1_0 :: bot:up
hole_top2_0 :: top
gen_bot:up3_0 :: Nat → bot:up

Lemmas:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_bot:up3_0(0) ⇔ bot
gen_bot:up3_0(+(x, 1)) ⇔ up(gen_bot:up3_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sent(gen_bot:up3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(34) BOUNDS(n^1, INF)