### (0) Obligation:

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

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
times(s(Y5_0), s(Y)) →+ plus(s(Y5_0), plus(Y, times(Y5_0, Y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [Y5_0 / s(Y5_0), Y / s(Y)].
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:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

Types:
plus :: s → s → s
times :: s → s → s
s :: s → s
hole_s1_0 :: s
gen_s2_0 :: Nat → s

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

Heuristically decided to analyse the following defined symbols:
plus, times

They will be analysed ascendingly in the following order:
plus < times

### (8) Obligation:

TRS:
Rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

Types:
plus :: s → s → s
times :: s → s → s
s :: s → s
hole_s1_0 :: s
gen_s2_0 :: Nat → s

Generator Equations:
gen_s2_0(0) ⇔ hole_s1_0
gen_s2_0(+(x, 1)) ⇔ s(gen_s2_0(x))

The following defined symbols remain to be analysed:
plus, times

They will be analysed ascendingly in the following order:
plus < times

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

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

### (10) Obligation:

TRS:
Rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

Types:
plus :: s → s → s
times :: s → s → s
s :: s → s
hole_s1_0 :: s
gen_s2_0 :: Nat → s

Generator Equations:
gen_s2_0(0) ⇔ hole_s1_0
gen_s2_0(+(x, 1)) ⇔ s(gen_s2_0(x))

The following defined symbols remain to be analysed:
times

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

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

### (12) Obligation:

TRS:
Rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

Types:
plus :: s → s → s
times :: s → s → s
s :: s → s
hole_s1_0 :: s
gen_s2_0 :: Nat → s

Generator Equations:
gen_s2_0(0) ⇔ hole_s1_0
gen_s2_0(+(x, 1)) ⇔ s(gen_s2_0(x))

No more defined symbols left to analyse.