### (0) Obligation:

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

t(N) → cs(r(q(N)), nt(ns(N)))
q(0) → 0
q(s(X)) → s(p(q(X), d(X)))
d(0) → 0
d(s(X)) → s(s(d(X)))
p(0, X) → X
p(X, 0) → X
p(s(X), s(Y)) → s(s(p(X, Y)))
f(0, X) → nil
f(s(X), cs(Y, Z)) → cs(Y, nf(X, a(Z)))
t(X) → nt(X)
s(X) → ns(X)
f(X1, X2) → nf(X1, X2)
a(nt(X)) → t(a(X))
a(ns(X)) → s(a(X))
a(nf(X1, X2)) → f(a(X1), a(X2))
a(X) → X

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
a(nt(X)) →+ cs(r(q(a(X))), nt(ns(a(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X / nt(X)].
The result substitution is [ ].

The rewrite sequence
a(nt(X)) →+ cs(r(q(a(X))), nt(ns(a(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / nt(X)].
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:

t(N) → cs(r(q(N)), nt(ns(N)))
q(0') → 0'
q(s(X)) → s(p(q(X), d(X)))
d(0') → 0'
d(s(X)) → s(s(d(X)))
p(0', X) → X
p(X, 0') → X
p(s(X), s(Y)) → s(s(p(X, Y)))
f(0', X) → nil
f(s(X), cs(Y, Z)) → cs(Y, nf(X, a(Z)))
t(X) → nt(X)
s(X) → ns(X)
f(X1, X2) → nf(X1, X2)
a(nt(X)) → t(a(X))
a(ns(X)) → s(a(X))
a(nf(X1, X2)) → f(a(X1), a(X2))
a(X) → X

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
t(N) → cs(r(q(N)), nt(ns(N)))
q(0') → 0'
q(s(X)) → s(p(q(X), d(X)))
d(0') → 0'
d(s(X)) → s(s(d(X)))
p(0', X) → X
p(X, 0') → X
p(s(X), s(Y)) → s(s(p(X, Y)))
f(0', X) → nil
f(s(X), cs(Y, Z)) → cs(Y, nf(X, a(Z)))
t(X) → nt(X)
s(X) → ns(X)
f(X1, X2) → nf(X1, X2)
a(nt(X)) → t(a(X))
a(ns(X)) → s(a(X))
a(nf(X1, X2)) → f(a(X1), a(X2))
a(X) → X

Types:
t :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
cs :: r → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
r :: ns:nt:cs:0':nil:nf → r
q :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nt :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
ns :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
0' :: ns:nt:cs:0':nil:nf
s :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
p :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
d :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
f :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nil :: ns:nt:cs:0':nil:nf
nf :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
a :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
hole_ns:nt:cs:0':nil:nf1_0 :: ns:nt:cs:0':nil:nf
hole_r2_0 :: r
gen_ns:nt:cs:0':nil:nf3_0 :: Nat → ns:nt:cs:0':nil:nf

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

Heuristically decided to analyse the following defined symbols:
q, p, d, a

They will be analysed ascendingly in the following order:
p < q
d < q

### (8) Obligation:

TRS:
Rules:
t(N) → cs(r(q(N)), nt(ns(N)))
q(0') → 0'
q(s(X)) → s(p(q(X), d(X)))
d(0') → 0'
d(s(X)) → s(s(d(X)))
p(0', X) → X
p(X, 0') → X
p(s(X), s(Y)) → s(s(p(X, Y)))
f(0', X) → nil
f(s(X), cs(Y, Z)) → cs(Y, nf(X, a(Z)))
t(X) → nt(X)
s(X) → ns(X)
f(X1, X2) → nf(X1, X2)
a(nt(X)) → t(a(X))
a(ns(X)) → s(a(X))
a(nf(X1, X2)) → f(a(X1), a(X2))
a(X) → X

Types:
t :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
cs :: r → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
r :: ns:nt:cs:0':nil:nf → r
q :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nt :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
ns :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
0' :: ns:nt:cs:0':nil:nf
s :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
p :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
d :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
f :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nil :: ns:nt:cs:0':nil:nf
nf :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
a :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
hole_ns:nt:cs:0':nil:nf1_0 :: ns:nt:cs:0':nil:nf
hole_r2_0 :: r
gen_ns:nt:cs:0':nil:nf3_0 :: Nat → ns:nt:cs:0':nil:nf

Generator Equations:
gen_ns:nt:cs:0':nil:nf3_0(0) ⇔ 0'
gen_ns:nt:cs:0':nil:nf3_0(+(x, 1)) ⇔ cs(r(0'), gen_ns:nt:cs:0':nil:nf3_0(x))

The following defined symbols remain to be analysed:
p, q, d, a

They will be analysed ascendingly in the following order:
p < q
d < q

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

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

### (10) Obligation:

TRS:
Rules:
t(N) → cs(r(q(N)), nt(ns(N)))
q(0') → 0'
q(s(X)) → s(p(q(X), d(X)))
d(0') → 0'
d(s(X)) → s(s(d(X)))
p(0', X) → X
p(X, 0') → X
p(s(X), s(Y)) → s(s(p(X, Y)))
f(0', X) → nil
f(s(X), cs(Y, Z)) → cs(Y, nf(X, a(Z)))
t(X) → nt(X)
s(X) → ns(X)
f(X1, X2) → nf(X1, X2)
a(nt(X)) → t(a(X))
a(ns(X)) → s(a(X))
a(nf(X1, X2)) → f(a(X1), a(X2))
a(X) → X

Types:
t :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
cs :: r → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
r :: ns:nt:cs:0':nil:nf → r
q :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nt :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
ns :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
0' :: ns:nt:cs:0':nil:nf
s :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
p :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
d :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
f :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nil :: ns:nt:cs:0':nil:nf
nf :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
a :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
hole_ns:nt:cs:0':nil:nf1_0 :: ns:nt:cs:0':nil:nf
hole_r2_0 :: r
gen_ns:nt:cs:0':nil:nf3_0 :: Nat → ns:nt:cs:0':nil:nf

Generator Equations:
gen_ns:nt:cs:0':nil:nf3_0(0) ⇔ 0'
gen_ns:nt:cs:0':nil:nf3_0(+(x, 1)) ⇔ cs(r(0'), gen_ns:nt:cs:0':nil:nf3_0(x))

The following defined symbols remain to be analysed:
d, q, a

They will be analysed ascendingly in the following order:
d < q

### (11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (12) Obligation:

TRS:
Rules:
t(N) → cs(r(q(N)), nt(ns(N)))
q(0') → 0'
q(s(X)) → s(p(q(X), d(X)))
d(0') → 0'
d(s(X)) → s(s(d(X)))
p(0', X) → X
p(X, 0') → X
p(s(X), s(Y)) → s(s(p(X, Y)))
f(0', X) → nil
f(s(X), cs(Y, Z)) → cs(Y, nf(X, a(Z)))
t(X) → nt(X)
s(X) → ns(X)
f(X1, X2) → nf(X1, X2)
a(nt(X)) → t(a(X))
a(ns(X)) → s(a(X))
a(nf(X1, X2)) → f(a(X1), a(X2))
a(X) → X

Types:
t :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
cs :: r → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
r :: ns:nt:cs:0':nil:nf → r
q :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nt :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
ns :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
0' :: ns:nt:cs:0':nil:nf
s :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
p :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
d :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
f :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nil :: ns:nt:cs:0':nil:nf
nf :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
a :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
hole_ns:nt:cs:0':nil:nf1_0 :: ns:nt:cs:0':nil:nf
hole_r2_0 :: r
gen_ns:nt:cs:0':nil:nf3_0 :: Nat → ns:nt:cs:0':nil:nf

Generator Equations:
gen_ns:nt:cs:0':nil:nf3_0(0) ⇔ 0'
gen_ns:nt:cs:0':nil:nf3_0(+(x, 1)) ⇔ cs(r(0'), gen_ns:nt:cs:0':nil:nf3_0(x))

The following defined symbols remain to be analysed:
q, a

### (13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (14) Obligation:

TRS:
Rules:
t(N) → cs(r(q(N)), nt(ns(N)))
q(0') → 0'
q(s(X)) → s(p(q(X), d(X)))
d(0') → 0'
d(s(X)) → s(s(d(X)))
p(0', X) → X
p(X, 0') → X
p(s(X), s(Y)) → s(s(p(X, Y)))
f(0', X) → nil
f(s(X), cs(Y, Z)) → cs(Y, nf(X, a(Z)))
t(X) → nt(X)
s(X) → ns(X)
f(X1, X2) → nf(X1, X2)
a(nt(X)) → t(a(X))
a(ns(X)) → s(a(X))
a(nf(X1, X2)) → f(a(X1), a(X2))
a(X) → X

Types:
t :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
cs :: r → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
r :: ns:nt:cs:0':nil:nf → r
q :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nt :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
ns :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
0' :: ns:nt:cs:0':nil:nf
s :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
p :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
d :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
f :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nil :: ns:nt:cs:0':nil:nf
nf :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
a :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
hole_ns:nt:cs:0':nil:nf1_0 :: ns:nt:cs:0':nil:nf
hole_r2_0 :: r
gen_ns:nt:cs:0':nil:nf3_0 :: Nat → ns:nt:cs:0':nil:nf

Generator Equations:
gen_ns:nt:cs:0':nil:nf3_0(0) ⇔ 0'
gen_ns:nt:cs:0':nil:nf3_0(+(x, 1)) ⇔ cs(r(0'), gen_ns:nt:cs:0':nil:nf3_0(x))

The following defined symbols remain to be analysed:
a

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

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

### (16) Obligation:

TRS:
Rules:
t(N) → cs(r(q(N)), nt(ns(N)))
q(0') → 0'
q(s(X)) → s(p(q(X), d(X)))
d(0') → 0'
d(s(X)) → s(s(d(X)))
p(0', X) → X
p(X, 0') → X
p(s(X), s(Y)) → s(s(p(X, Y)))
f(0', X) → nil
f(s(X), cs(Y, Z)) → cs(Y, nf(X, a(Z)))
t(X) → nt(X)
s(X) → ns(X)
f(X1, X2) → nf(X1, X2)
a(nt(X)) → t(a(X))
a(ns(X)) → s(a(X))
a(nf(X1, X2)) → f(a(X1), a(X2))
a(X) → X

Types:
t :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
cs :: r → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
r :: ns:nt:cs:0':nil:nf → r
q :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nt :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
ns :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
0' :: ns:nt:cs:0':nil:nf
s :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
p :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
d :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
f :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
nil :: ns:nt:cs:0':nil:nf
nf :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
a :: ns:nt:cs:0':nil:nf → ns:nt:cs:0':nil:nf
hole_ns:nt:cs:0':nil:nf1_0 :: ns:nt:cs:0':nil:nf
hole_r2_0 :: r
gen_ns:nt:cs:0':nil:nf3_0 :: Nat → ns:nt:cs:0':nil:nf

Generator Equations:
gen_ns:nt:cs:0':nil:nf3_0(0) ⇔ 0'
gen_ns:nt:cs:0':nil:nf3_0(+(x, 1)) ⇔ cs(r(0'), gen_ns:nt:cs:0':nil:nf3_0(x))

No more defined symbols left to analyse.