### (0) Obligation:

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

minus_active(0, y) → 0
mark(0) → 0
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0) → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0, s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0, s(y)) → 0
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(s(div(s(x2347_1), s(y3881_1)))) →+ s(if_active(ge_active(mark(x2347_1), y3881_1), s(div(minus(mark(x2347_1), y3881_1), s(y3881_1))), 0))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [x2347_1 / s(div(s(x2347_1), s(y3881_1)))].
The result substitution is [ ].

The rewrite sequence
mark(s(div(s(x2347_1), s(y3881_1)))) →+ s(if_active(ge_active(mark(x2347_1), y3881_1), s(div(minus(mark(x2347_1), y3881_1), s(y3881_1))), 0))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0,0].
The pumping substitution is [x2347_1 / s(div(s(x2347_1), s(y3881_1)))].
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:

minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat → 0':s:true:minus:false:ge:div:if

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

Heuristically decided to analyse the following defined symbols:
minus_active, mark, ge_active

They will be analysed ascendingly in the following order:
minus_active < mark
ge_active < mark

### (8) Obligation:

TRS:
Rules:
minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat → 0':s:true:minus:false:ge:div:if

Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) ⇔ 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:minus:false:ge:div:if2_0(x))

The following defined symbols remain to be analysed:
minus_active, mark, ge_active

They will be analysed ascendingly in the following order:
minus_active < mark
ge_active < mark

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

Proved the following rewrite lemma:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)

Induction Base:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(0), gen_0':s:true:minus:false:ge:div:if2_0(0)) →RΩ(1)
0'

Induction Step:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(+(n4_0, 1)), gen_0':s:true:minus:false:ge:div:if2_0(+(n4_0, 1))) →RΩ(1)
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) →IH
gen_0':s:true:minus:false:ge:div:if2_0(0)

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

### (11) Obligation:

TRS:
Rules:
minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat → 0':s:true:minus:false:ge:div:if

Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) ⇔ 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:minus:false:ge:div:if2_0(x))

The following defined symbols remain to be analysed:
ge_active, mark

They will be analysed ascendingly in the following order:
ge_active < mark

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

Proved the following rewrite lemma:
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n465_0), gen_0':s:true:minus:false:ge:div:if2_0(n465_0)) → true, rt ∈ Ω(1 + n4650)

Induction Base:
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(0), gen_0':s:true:minus:false:ge:div:if2_0(0)) →RΩ(1)
true

Induction Step:
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(+(n465_0, 1)), gen_0':s:true:minus:false:ge:div:if2_0(+(n465_0, 1))) →RΩ(1)
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n465_0), gen_0':s:true:minus:false:ge:div:if2_0(n465_0)) →IH
true

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

### (14) Obligation:

TRS:
Rules:
minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat → 0':s:true:minus:false:ge:div:if

Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n465_0), gen_0':s:true:minus:false:ge:div:if2_0(n465_0)) → true, rt ∈ Ω(1 + n4650)

Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) ⇔ 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:minus:false:ge:div:if2_0(x))

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:
mark(gen_0':s:true:minus:false:ge:div:if2_0(n1054_0)) → gen_0':s:true:minus:false:ge:div:if2_0(n1054_0), rt ∈ Ω(1 + n10540)

Induction Base:
mark(gen_0':s:true:minus:false:ge:div:if2_0(0)) →RΩ(1)
0'

Induction Step:
mark(gen_0':s:true:minus:false:ge:div:if2_0(+(n1054_0, 1))) →RΩ(1)
s(mark(gen_0':s:true:minus:false:ge:div:if2_0(n1054_0))) →IH
s(gen_0':s:true:minus:false:ge:div:if2_0(c1055_0))

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

### (17) Obligation:

TRS:
Rules:
minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat → 0':s:true:minus:false:ge:div:if

Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n465_0), gen_0':s:true:minus:false:ge:div:if2_0(n465_0)) → true, rt ∈ Ω(1 + n4650)
mark(gen_0':s:true:minus:false:ge:div:if2_0(n1054_0)) → gen_0':s:true:minus:false:ge:div:if2_0(n1054_0), rt ∈ Ω(1 + n10540)

Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) ⇔ 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:minus:false:ge:div:if2_0(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)

### (20) Obligation:

TRS:
Rules:
minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat → 0':s:true:minus:false:ge:div:if

Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n465_0), gen_0':s:true:minus:false:ge:div:if2_0(n465_0)) → true, rt ∈ Ω(1 + n4650)
mark(gen_0':s:true:minus:false:ge:div:if2_0(n1054_0)) → gen_0':s:true:minus:false:ge:div:if2_0(n1054_0), rt ∈ Ω(1 + n10540)

Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) ⇔ 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:minus:false:ge:div:if2_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)

### (23) Obligation:

TRS:
Rules:
minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat → 0':s:true:minus:false:ge:div:if

Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n465_0), gen_0':s:true:minus:false:ge:div:if2_0(n465_0)) → true, rt ∈ Ω(1 + n4650)

Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) ⇔ 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:minus:false:ge:div:if2_0(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)

### (26) Obligation:

TRS:
Rules:
minus_active(0', y) → 0'
mark(0') → 0'
minus_active(s(x), s(y)) → minus_active(x, y)
mark(s(x)) → s(mark(x))
ge_active(x, 0') → true
mark(minus(x, y)) → minus_active(x, y)
ge_active(0', s(y)) → false
mark(ge(x, y)) → ge_active(x, y)
ge_active(s(x), s(y)) → ge_active(x, y)
mark(div(x, y)) → div_active(mark(x), y)
div_active(0', s(y)) → 0'
mark(if(x, y, z)) → if_active(mark(x), y, z)
div_active(s(x), s(y)) → if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) → mark(x)
minus_active(x, y) → minus(x, y)
if_active(false, x, y) → mark(y)
ge_active(x, y) → ge(x, y)
if_active(x, y, z) → if(x, y, z)
div_active(x, y) → div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if → 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat → 0':s:true:minus:false:ge:div:if

Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) ⇔ 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:minus:false:ge:div:if2_0(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) → gen_0':s:true:minus:false:ge:div:if2_0(0), rt ∈ Ω(1 + n40)