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:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
activate(n__0) → 0
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(n¹):
The rewrite sequence
minus(n__s(X), n__s(Y)) →⁺ minus(X, Y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X / n__s(X), Y / n__s(Y)].
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:

minus(n__0, Y) → 0'
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0', n__s(Y)) → 0'
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0' → n__0
s(X) → n__s(X)
activate(n__0) → 0'
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:
minus(n__0, Y) → 0'
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0', n__s(Y)) → 0'
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0' → n__0
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(X) → X

Types:
minus :: n__0:n__s → n__0:n__s → n__0:n__s
n__0 :: n__0:n__s
0' :: n__0:n__s
n__s :: n__0:n__s → n__0:n__s
activate :: n__0:n__s → n__0:n__s
geq :: n__0:n__s → n__0:n__s → true:false
true :: true:false
false :: true:false
div :: n__0:n__s → n__0:n__s → n__0:n__s
s :: n__0:n__s → n__0:n__s
if :: true:false → n__0:n__s → n__0:n__s → n__0:n__s
hole_n__0:n__s1_1 :: n__0:n__s
hole_true:false2_1 :: true:false
gen_n__0:n__s3_1 :: Nat → n__0:n__s

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

Heuristically decided to analyse the following defined symbols:
minus, geq, div

They will be analysed ascendingly in the following order:
minus < div
geq < div

(8) Obligation:

TRS:
Rules:
minus(n__0, Y) → 0'
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0', n__s(Y)) → 0'
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0' → n__0
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(X) → X

Types:
minus :: n__0:n__s → n__0:n__s → n__0:n__s
n__0 :: n__0:n__s
0' :: n__0:n__s
n__s :: n__0:n__s → n__0:n__s
activate :: n__0:n__s → n__0:n__s
geq :: n__0:n__s → n__0:n__s → true:false
true :: true:false
false :: true:false
div :: n__0:n__s → n__0:n__s → n__0:n__s
s :: n__0:n__s → n__0:n__s
if :: true:false → n__0:n__s → n__0:n__s → n__0:n__s
hole_n__0:n__s1_1 :: n__0:n__s
hole_true:false2_1 :: true:false
gen_n__0:n__s3_1 :: Nat → n__0:n__s

Generator Equations:
gen_n__0:n__s3_1(0) ⇔ n__0
gen_n__0:n__s3_1(+(x, 1)) ⇔ n__s(gen_n__0:n__s3_1(x))

The following defined symbols remain to be analysed:
minus, geq, div

They will be analysed ascendingly in the following order:
minus < div
geq < div

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)

Induction Base:
minus(gen_n__0:n__s3_1(+(2, 0)), gen_n__0:n__s3_1(+(1, 0))) →_R^Ω(1)
minus(activate(gen_n__0:n__s3_1(1)), activate(gen_n__0:n__s3_1(0))) →_R^Ω(1)
minus(gen_n__0:n__s3_1(1), activate(gen_n__0:n__s3_1(0))) →_R^Ω(1)
minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0))

Induction Step:
minus(gen_n__0:n__s3_1(+(2, +(n5_1, 1))), gen_n__0:n__s3_1(+(1, +(n5_1, 1)))) →_R^Ω(1)
minus(activate(gen_n__0:n__s3_1(+(2, n5_1))), activate(gen_n__0:n__s3_1(+(1, n5_1)))) →_R^Ω(1)
minus(gen_n__0:n__s3_1(+(2, n5_1)), activate(gen_n__0:n__s3_1(+(1, n5_1)))) →_R^Ω(1)
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) →_IH
minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
minus(n__0, Y) → 0'
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0', n__s(Y)) → 0'
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0' → n__0
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(X) → X

Types:
minus :: n__0:n__s → n__0:n__s → n__0:n__s
n__0 :: n__0:n__s
0' :: n__0:n__s
n__s :: n__0:n__s → n__0:n__s
activate :: n__0:n__s → n__0:n__s
geq :: n__0:n__s → n__0:n__s → true:false
true :: true:false
false :: true:false
div :: n__0:n__s → n__0:n__s → n__0:n__s
s :: n__0:n__s → n__0:n__s
if :: true:false → n__0:n__s → n__0:n__s → n__0:n__s
hole_n__0:n__s1_1 :: n__0:n__s
hole_true:false2_1 :: true:false
gen_n__0:n__s3_1 :: Nat → n__0:n__s

Lemmas:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)

Generator Equations:
gen_n__0:n__s3_1(0) ⇔ n__0
gen_n__0:n__s3_1(+(x, 1)) ⇔ n__s(gen_n__0:n__s3_1(x))

The following defined symbols remain to be analysed:
geq, div

They will be analysed ascendingly in the following order:
geq < div

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

Proved the following rewrite lemma:
geq(gen_n__0:n__s3_1(n2522_1), gen_n__0:n__s3_1(+(1, n2522_1))) → false, rt ∈ Ω(1 + n2522₁)

Induction Base:
geq(gen_n__0:n__s3_1(0), gen_n__0:n__s3_1(+(1, 0))) →_R^Ω(1)
false

Induction Step:
geq(gen_n__0:n__s3_1(+(n2522_1, 1)), gen_n__0:n__s3_1(+(1, +(n2522_1, 1)))) →_R^Ω(1)
geq(activate(gen_n__0:n__s3_1(n2522_1)), activate(gen_n__0:n__s3_1(+(1, n2522_1)))) →_R^Ω(1)
geq(gen_n__0:n__s3_1(n2522_1), activate(gen_n__0:n__s3_1(+(1, n2522_1)))) →_R^Ω(1)
geq(gen_n__0:n__s3_1(n2522_1), gen_n__0:n__s3_1(+(1, n2522_1))) →_IH
false

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
minus(n__0, Y) → 0'
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0', n__s(Y)) → 0'
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0' → n__0
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(X) → X

Types:
minus :: n__0:n__s → n__0:n__s → n__0:n__s
n__0 :: n__0:n__s
0' :: n__0:n__s
n__s :: n__0:n__s → n__0:n__s
activate :: n__0:n__s → n__0:n__s
geq :: n__0:n__s → n__0:n__s → true:false
true :: true:false
false :: true:false
div :: n__0:n__s → n__0:n__s → n__0:n__s
s :: n__0:n__s → n__0:n__s
if :: true:false → n__0:n__s → n__0:n__s → n__0:n__s
hole_n__0:n__s1_1 :: n__0:n__s
hole_true:false2_1 :: true:false
gen_n__0:n__s3_1 :: Nat → n__0:n__s

Lemmas:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)
geq(gen_n__0:n__s3_1(n2522_1), gen_n__0:n__s3_1(+(1, n2522_1))) → false, rt ∈ Ω(1 + n2522₁)

Generator Equations:
gen_n__0:n__s3_1(0) ⇔ n__0
gen_n__0:n__s3_1(+(x, 1)) ⇔ n__s(gen_n__0:n__s3_1(x))

The following defined symbols remain to be analysed:
div

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
minus(n__0, Y) → 0'
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0', n__s(Y)) → 0'
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0' → n__0
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(X) → X

Types:
minus :: n__0:n__s → n__0:n__s → n__0:n__s
n__0 :: n__0:n__s
0' :: n__0:n__s
n__s :: n__0:n__s → n__0:n__s
activate :: n__0:n__s → n__0:n__s
geq :: n__0:n__s → n__0:n__s → true:false
true :: true:false
false :: true:false
div :: n__0:n__s → n__0:n__s → n__0:n__s
s :: n__0:n__s → n__0:n__s
if :: true:false → n__0:n__s → n__0:n__s → n__0:n__s
hole_n__0:n__s1_1 :: n__0:n__s
hole_true:false2_1 :: true:false
gen_n__0:n__s3_1 :: Nat → n__0:n__s

Lemmas:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)
geq(gen_n__0:n__s3_1(n2522_1), gen_n__0:n__s3_1(+(1, n2522_1))) → false, rt ∈ Ω(1 + n2522₁)

Generator Equations:
gen_n__0:n__s3_1(0) ⇔ n__0
gen_n__0:n__s3_1(+(x, 1)) ⇔ n__s(gen_n__0:n__s3_1(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)

(18) BOUNDS(n^1, INF)

(19) Obligation:

TRS:
Rules:
minus(n__0, Y) → 0'
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0', n__s(Y)) → 0'
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0' → n__0
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(X) → X

Types:
minus :: n__0:n__s → n__0:n__s → n__0:n__s
n__0 :: n__0:n__s
0' :: n__0:n__s
n__s :: n__0:n__s → n__0:n__s
activate :: n__0:n__s → n__0:n__s
geq :: n__0:n__s → n__0:n__s → true:false
true :: true:false
false :: true:false
div :: n__0:n__s → n__0:n__s → n__0:n__s
s :: n__0:n__s → n__0:n__s
if :: true:false → n__0:n__s → n__0:n__s → n__0:n__s
hole_n__0:n__s1_1 :: n__0:n__s
hole_true:false2_1 :: true:false
gen_n__0:n__s3_1 :: Nat → n__0:n__s

Lemmas:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)
geq(gen_n__0:n__s3_1(n2522_1), gen_n__0:n__s3_1(+(1, n2522_1))) → false, rt ∈ Ω(1 + n2522₁)

Generator Equations:
gen_n__0:n__s3_1(0) ⇔ n__0
gen_n__0:n__s3_1(+(x, 1)) ⇔ n__s(gen_n__0:n__s3_1(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
minus(n__0, Y) → 0'
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0', n__s(Y)) → 0'
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0' → n__0
s(X) → n__s(X)
activate(n__0) → 0'
activate(n__s(X)) → s(X)
activate(X) → X

Types:
minus :: n__0:n__s → n__0:n__s → n__0:n__s
n__0 :: n__0:n__s
0' :: n__0:n__s
n__s :: n__0:n__s → n__0:n__s
activate :: n__0:n__s → n__0:n__s
geq :: n__0:n__s → n__0:n__s → true:false
true :: true:false
false :: true:false
div :: n__0:n__s → n__0:n__s → n__0:n__s
s :: n__0:n__s → n__0:n__s
if :: true:false → n__0:n__s → n__0:n__s → n__0:n__s
hole_n__0:n__s1_1 :: n__0:n__s
hole_true:false2_1 :: true:false
gen_n__0:n__s3_1 :: Nat → n__0:n__s

Lemmas:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)

Generator Equations:
gen_n__0:n__s3_1(0) ⇔ n__0
gen_n__0:n__s3_1(+(x, 1)) ⇔ n__s(gen_n__0:n__s3_1(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_n__0:n__s3_1(+(2, n5_1)), gen_n__0:n__s3_1(+(1, n5_1))) → minus(gen_n__0:n__s3_1(1), gen_n__0:n__s3_1(0)), rt ∈ Ω(1 + n5₁)

(24) BOUNDS(n^1, INF)