### (0) Obligation:

Runtime Complexity TRS:

The TRS R consists of the following rules:

f(s(x), y) → f(x, s(x))

f(x, s(y)) → f(y, 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(s(x), y) →^{+} f(x, s(x))

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(x)].

### (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(s(x), y) → f(x, s(x))

f(x, s(y)) → f(y, x)

S is empty.

Rewrite Strategy: FULL

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

Infered types.

### (6) Obligation:

TRS:

Rules:

f(s(x), y) → f(x, s(x))

f(x, s(y)) → f(y, x)

Types:

f :: s → s → f

s :: s → s

hole_f1_0 :: f

hole_s2_0 :: s

gen_s3_0 :: Nat → s

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

Heuristically decided to analyse the following defined symbols:

f
### (8) Obligation:

TRS:

Rules:

f(

s(

x),

y) →

f(

x,

s(

x))

f(

x,

s(

y)) →

f(

y,

x)

Types:

f :: s → s → f

s :: s → s

hole_f1_0 :: f

hole_s2_0 :: s

gen_s3_0 :: Nat → s

Generator Equations:

gen_s3_0(0) ⇔ hole_s2_0

gen_s3_0(+(x, 1)) ⇔ s(gen_s3_0(x))

The following defined symbols remain to be analysed:

f

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

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

TRS:

Rules:

f(

s(

x),

y) →

f(

x,

s(

x))

f(

x,

s(

y)) →

f(

y,

x)

Types:

f :: s → s → f

s :: s → s

hole_f1_0 :: f

hole_s2_0 :: s

gen_s3_0 :: Nat → s

Generator Equations:

gen_s3_0(0) ⇔ hole_s2_0

gen_s3_0(+(x, 1)) ⇔ s(gen_s3_0(x))

No more defined symbols left to analyse.