### (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))
x(N, 0) → 0
x(N, s(M)) → plus(x(N, M), N)
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
plus(N, s(M)) →+ s(plus(N, M))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [M / s(M)].
The result substitution is [ ].

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

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

Heuristically decided to analyse the following defined symbols:
plus, x

They will be analysed ascendingly in the following order:
plus < x

### (8) Obligation:

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

They will be analysed ascendingly in the following order:
plus < x

### (9) 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)) →RΩ(1)
gen_0':s4_0(a)

Induction Step:
plus(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) →RΩ(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 ircR ∈ Ω(n).

### (11) Obligation:

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

The following defined symbols remain to be analysed:
x

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

Proved the following rewrite lemma:
x(gen_0':s4_0(a), gen_0':s4_0(n441_0)) → gen_0':s4_0(*(n441_0, a)), rt ∈ Ω(1 + a·n4410 + n4410)

Induction Base:
x(gen_0':s4_0(a), gen_0':s4_0(0)) →RΩ(1)
0'

Induction Step:
x(gen_0':s4_0(a), gen_0':s4_0(+(n441_0, 1))) →RΩ(1)
plus(x(gen_0':s4_0(a), gen_0':s4_0(n441_0)), gen_0':s4_0(a)) →IH
plus(gen_0':s4_0(*(c442_0, a)), gen_0':s4_0(a)) →LΩ(1 + a)
gen_0':s4_0(+(a, *(n441_0, a)))

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

### (14) Obligation:

TRS:
Rules:
and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
x(N, 0') → 0'
x(N, s(M)) → plus(x(N, M), N)
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
x :: 0':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)
x(gen_0':s4_0(a), gen_0':s4_0(n441_0)) → gen_0':s4_0(*(n441_0, a)), rt ∈ Ω(1 + a·n4410 + n4410)

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.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
x(gen_0':s4_0(a), gen_0':s4_0(n441_0)) → gen_0':s4_0(*(n441_0, a)), rt ∈ Ω(1 + a·n4410 + n4410)

### (17) Obligation:

TRS:
Rules:
and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
x(N, 0') → 0'
x(N, s(M)) → plus(x(N, M), N)
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
x :: 0':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)
x(gen_0':s4_0(a), gen_0':s4_0(n441_0)) → gen_0':s4_0(*(n441_0, a)), rt ∈ Ω(1 + a·n4410 + n4410)

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.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
x(gen_0':s4_0(a), gen_0':s4_0(n441_0)) → gen_0':s4_0(*(n441_0, a)), rt ∈ Ω(1 + a·n4410 + n4410)

### (20) Obligation:

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

### (21) 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)