The Runtime Complexity (full) 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:

and(tt, X) → activate(X)
plus(N, 0) → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X

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:

and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X

Types:
and :: tt → and:activate → and:activate
tt :: tt
activate :: and:activate → and:activate
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
plus

(6) Obligation:

TRS:
Rules:
and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X

Types:
and :: tt → and:activate → and:activate
tt :: tt
activate :: and:activate → and:activate
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
plus

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

Proved the following rewrite lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n6₀)

Induction Base:
plus(gen_0':s4_0(a), gen_0':s4_0(0)) →_R^Ω(1)
gen_0':s4_0(a)

Induction Step:
plus(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) →_R^Ω(1)
s(plus(gen_0':s4_0(a), gen_0':s4_0(n6_0))) →_IH
s(gen_0':s4_0(+(a, c7_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X

Types:
and :: tt → and:activate → and:activate
tt :: tt
activate :: and:activate → and:activate
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n6₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

TRS:
Rules:
and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X

Types:
and :: tt → and:activate → and:activate
tt :: tt
activate :: and:activate → and:activate
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n6₀)

(14) BOUNDS(n^1, INF)