### (0) Obligation:

Runtime Complexity TRS:

The TRS R consists of the following rules:

f(c(s(x), y)) → f(c(x, s(y)))

g(c(x, s(y))) → g(c(s(x), y))

g(s(f(x))) → g(f(x))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n^{1}):

The rewrite sequence

f(c(s(x), y)) →^{+} f(c(x, s(y)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x / s(x)].

The result substitution is [y / s(y)].

### (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:

f(c(s(x), y)) → f(c(x, s(y)))

g(c(x, s(y))) → g(c(s(x), y))

g(s(f(x))) → g(f(x))

S is empty.

Rewrite Strategy: FULL

### (5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

### (6) Obligation:

TRS:

Rules:

f(c(s(x), y)) → f(c(x, s(y)))

g(c(x, s(y))) → g(c(s(x), y))

g(s(f(x))) → g(f(x))

Types:

f :: s:c → s:c

c :: s:c → s:c → s:c

s :: s:c → s:c

g :: s:c → g

hole_s:c1_0 :: s:c

hole_g2_0 :: g

gen_s:c3_0 :: Nat → s:c

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

Heuristically decided to analyse the following defined symbols:

f,

gThey will be analysed ascendingly in the following order:

f < g

### (8) Obligation:

TRS:

Rules:

f(

c(

s(

x),

y)) →

f(

c(

x,

s(

y)))

g(

c(

x,

s(

y))) →

g(

c(

s(

x),

y))

g(

s(

f(

x))) →

g(

f(

x))

Types:

f :: s:c → s:c

c :: s:c → s:c → s:c

s :: s:c → s:c

g :: s:c → g

hole_s:c1_0 :: s:c

hole_g2_0 :: g

gen_s:c3_0 :: Nat → s:c

Generator Equations:

gen_s:c3_0(0) ⇔ hole_s:c1_0

gen_s:c3_0(+(x, 1)) ⇔ c(hole_s:c1_0, gen_s:c3_0(x))

The following defined symbols remain to be analysed:

f, g

They will be analysed ascendingly in the following order:

f < g

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

Could not prove a rewrite lemma for the defined symbol f.
### (10) Obligation:

TRS:

Rules:

f(

c(

s(

x),

y)) →

f(

c(

x,

s(

y)))

g(

c(

x,

s(

y))) →

g(

c(

s(

x),

y))

g(

s(

f(

x))) →

g(

f(

x))

Types:

f :: s:c → s:c

c :: s:c → s:c → s:c

s :: s:c → s:c

g :: s:c → g

hole_s:c1_0 :: s:c

hole_g2_0 :: g

gen_s:c3_0 :: Nat → s:c

Generator Equations:

gen_s:c3_0(0) ⇔ hole_s:c1_0

gen_s:c3_0(+(x, 1)) ⇔ c(hole_s:c1_0, gen_s:c3_0(x))

The following defined symbols remain to be analysed:

g

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

Could not prove a rewrite lemma for the defined symbol g.
### (12) Obligation:

TRS:

Rules:

f(

c(

s(

x),

y)) →

f(

c(

x,

s(

y)))

g(

c(

x,

s(

y))) →

g(

c(

s(

x),

y))

g(

s(

f(

x))) →

g(

f(

x))

Types:

f :: s:c → s:c

c :: s:c → s:c → s:c

s :: s:c → s:c

g :: s:c → g

hole_s:c1_0 :: s:c

hole_g2_0 :: g

gen_s:c3_0 :: Nat → s:c

Generator Equations:

gen_s:c3_0(0) ⇔ hole_s:c1_0

gen_s:c3_0(+(x, 1)) ⇔ c(hole_s:c1_0, gen_s:c3_0(x))

No more defined symbols left to analyse.