### (0) Obligation:

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

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
D(+(x, y)) →+ +(D(x), D(y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / +(x, 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:

D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

Types:
D :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
t :: t:1':constant:0':+':*':-
1' :: t:1':constant:0':+':*':-
constant :: t:1':constant:0':+':*':-
0' :: t:1':constant:0':+':*':-
+' :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
*' :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
- :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
hole_t:1':constant:0':+':*':-1_0 :: t:1':constant:0':+':*':-
gen_t:1':constant:0':+':*':-2_0 :: Nat → t:1':constant:0':+':*':-

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

Heuristically decided to analyse the following defined symbols:
D

### (8) Obligation:

TRS:
Rules:
D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

Types:
D :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
t :: t:1':constant:0':+':*':-
1' :: t:1':constant:0':+':*':-
constant :: t:1':constant:0':+':*':-
0' :: t:1':constant:0':+':*':-
+' :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
*' :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
- :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
hole_t:1':constant:0':+':*':-1_0 :: t:1':constant:0':+':*':-
gen_t:1':constant:0':+':*':-2_0 :: Nat → t:1':constant:0':+':*':-

Generator Equations:
gen_t:1':constant:0':+':*':-2_0(0) ⇔ t
gen_t:1':constant:0':+':*':-2_0(+(x, 1)) ⇔ +'(t, gen_t:1':constant:0':+':*':-2_0(x))

The following defined symbols remain to be analysed:
D

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

Proved the following rewrite lemma:
D(gen_t:1':constant:0':+':*':-2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Induction Base:
D(gen_t:1':constant:0':+':*':-2_0(0))

Induction Step:
D(gen_t:1':constant:0':+':*':-2_0(+(n4_0, 1))) →RΩ(1)
+'(D(t), D(gen_t:1':constant:0':+':*':-2_0(n4_0))) →RΩ(1)
+'(1', D(gen_t:1':constant:0':+':*':-2_0(n4_0))) →IH
+'(1', *3_0)

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

### (11) Obligation:

TRS:
Rules:
D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

Types:
D :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
t :: t:1':constant:0':+':*':-
1' :: t:1':constant:0':+':*':-
constant :: t:1':constant:0':+':*':-
0' :: t:1':constant:0':+':*':-
+' :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
*' :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
- :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
hole_t:1':constant:0':+':*':-1_0 :: t:1':constant:0':+':*':-
gen_t:1':constant:0':+':*':-2_0 :: Nat → t:1':constant:0':+':*':-

Lemmas:
D(gen_t:1':constant:0':+':*':-2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_t:1':constant:0':+':*':-2_0(0) ⇔ t
gen_t:1':constant:0':+':*':-2_0(+(x, 1)) ⇔ +'(t, gen_t:1':constant:0':+':*':-2_0(x))

No more defined symbols left to analyse.

### (12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
D(gen_t:1':constant:0':+':*':-2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

### (14) Obligation:

TRS:
Rules:
D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

Types:
D :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
t :: t:1':constant:0':+':*':-
1' :: t:1':constant:0':+':*':-
constant :: t:1':constant:0':+':*':-
0' :: t:1':constant:0':+':*':-
+' :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
*' :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
- :: t:1':constant:0':+':*':- → t:1':constant:0':+':*':- → t:1':constant:0':+':*':-
hole_t:1':constant:0':+':*':-1_0 :: t:1':constant:0':+':*':-
gen_t:1':constant:0':+':*':-2_0 :: Nat → t:1':constant:0':+':*':-

Lemmas:
D(gen_t:1':constant:0':+':*':-2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_t:1':constant:0':+':*':-2_0(0) ⇔ t
gen_t:1':constant:0':+':*':-2_0(+(x, 1)) ⇔ +'(t, gen_t:1':constant:0':+':*':-2_0(x))

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
D(gen_t:1':constant:0':+':*':-2_0(n4_0)) → *3_0, rt ∈ Ω(n40)