### (0) Obligation:

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

0(#) → #
+(#, x) → x
+(x, #) → x
+(0(x), 0(y)) → 0(+(x, y))
+(0(x), 1(y)) → 1(+(x, y))
+(1(x), 0(y)) → 1(+(x, y))
+(0(x), j(y)) → j(+(x, y))
+(j(x), 0(y)) → j(+(x, y))
+(1(x), 1(y)) → j(+(+(x, y), 1(#)))
+(j(x), j(y)) → 1(+(+(x, y), j(#)))
+(1(x), j(y)) → 0(+(x, y))
+(j(x), 1(y)) → 0(+(x, y))
+(+(x, y), z) → +(x, +(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +(x, opp(y))
*(#, x) → #
*(0(x), y) → 0(*(x, y))
*(1(x), y) → +(0(*(x, y)), y)
*(j(x), y) → -(0(*(x, y)), y)
*(*(x, y), z) → *(x, *(y, z))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
+(1(x), 1(y)) →+ j(+(+(x, y), 1(#)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [x / 1(x), y / 1(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:

0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

Types:
0 :: #:1:j → #:1:j
# :: #:1:j
+' :: #:1:j → #:1:j → #:1:j
1 :: #:1:j → #:1:j
j :: #:1:j → #:1:j
opp :: #:1:j → #:1:j
- :: #:1:j → #:1:j → #:1:j
*' :: #:1:j → #:1:j → #:1:j
hole_#:1:j1_2 :: #:1:j
gen_#:1:j2_2 :: Nat → #:1:j

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

Heuristically decided to analyse the following defined symbols:
+', opp, *'

They will be analysed ascendingly in the following order:
+' < *'

### (8) Obligation:

TRS:
Rules:
0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

Types:
0 :: #:1:j → #:1:j
# :: #:1:j
+' :: #:1:j → #:1:j → #:1:j
1 :: #:1:j → #:1:j
j :: #:1:j → #:1:j
opp :: #:1:j → #:1:j
- :: #:1:j → #:1:j → #:1:j
*' :: #:1:j → #:1:j → #:1:j
hole_#:1:j1_2 :: #:1:j
gen_#:1:j2_2 :: Nat → #:1:j

Generator Equations:
gen_#:1:j2_2(0) ⇔ #
gen_#:1:j2_2(+(x, 1)) ⇔ 1(gen_#:1:j2_2(x))

The following defined symbols remain to be analysed:
+', opp, *'

They will be analysed ascendingly in the following order:
+' < *'

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

Proved the following rewrite lemma:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)

Induction Base:
+'(gen_#:1:j2_2(+(1, 0)), gen_#:1:j2_2(+(1, 0)))

Induction Step:
+'(gen_#:1:j2_2(+(1, +(n4_2, 1))), gen_#:1:j2_2(+(1, +(n4_2, 1)))) →RΩ(1)
j(+'(+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))), 1(#))) →IH
j(+'(*3_2, 1(#)))

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

### (11) Obligation:

TRS:
Rules:
0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

Types:
0 :: #:1:j → #:1:j
# :: #:1:j
+' :: #:1:j → #:1:j → #:1:j
1 :: #:1:j → #:1:j
j :: #:1:j → #:1:j
opp :: #:1:j → #:1:j
- :: #:1:j → #:1:j → #:1:j
*' :: #:1:j → #:1:j → #:1:j
hole_#:1:j1_2 :: #:1:j
gen_#:1:j2_2 :: Nat → #:1:j

Lemmas:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)

Generator Equations:
gen_#:1:j2_2(0) ⇔ #
gen_#:1:j2_2(+(x, 1)) ⇔ 1(gen_#:1:j2_2(x))

The following defined symbols remain to be analysed:
opp, *'

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

Proved the following rewrite lemma:
opp(gen_#:1:j2_2(+(1, n6451040_2))) → *3_2, rt ∈ Ω(n64510402)

Induction Base:
opp(gen_#:1:j2_2(+(1, 0)))

Induction Step:
opp(gen_#:1:j2_2(+(1, +(n6451040_2, 1)))) →RΩ(1)
j(opp(gen_#:1:j2_2(+(1, n6451040_2)))) →IH
j(*3_2)

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

### (14) Obligation:

TRS:
Rules:
0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

Types:
0 :: #:1:j → #:1:j
# :: #:1:j
+' :: #:1:j → #:1:j → #:1:j
1 :: #:1:j → #:1:j
j :: #:1:j → #:1:j
opp :: #:1:j → #:1:j
- :: #:1:j → #:1:j → #:1:j
*' :: #:1:j → #:1:j → #:1:j
hole_#:1:j1_2 :: #:1:j
gen_#:1:j2_2 :: Nat → #:1:j

Lemmas:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)
opp(gen_#:1:j2_2(+(1, n6451040_2))) → *3_2, rt ∈ Ω(n64510402)

Generator Equations:
gen_#:1:j2_2(0) ⇔ #
gen_#:1:j2_2(+(x, 1)) ⇔ 1(gen_#:1:j2_2(x))

The following defined symbols remain to be analysed:
*'

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

Proved the following rewrite lemma:
*'(gen_#:1:j2_2(n6454175_2), gen_#:1:j2_2(0)) → gen_#:1:j2_2(0), rt ∈ Ω(1 + n64541752)

Induction Base:
*'(gen_#:1:j2_2(0), gen_#:1:j2_2(0)) →RΩ(1)
#

Induction Step:
*'(gen_#:1:j2_2(+(n6454175_2, 1)), gen_#:1:j2_2(0)) →RΩ(1)
+'(0(*'(gen_#:1:j2_2(n6454175_2), gen_#:1:j2_2(0))), gen_#:1:j2_2(0)) →IH
+'(0(gen_#:1:j2_2(0)), gen_#:1:j2_2(0)) →RΩ(1)
+'(#, gen_#:1:j2_2(0)) →RΩ(1)
gen_#:1:j2_2(0)

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

### (17) Obligation:

TRS:
Rules:
0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

Types:
0 :: #:1:j → #:1:j
# :: #:1:j
+' :: #:1:j → #:1:j → #:1:j
1 :: #:1:j → #:1:j
j :: #:1:j → #:1:j
opp :: #:1:j → #:1:j
- :: #:1:j → #:1:j → #:1:j
*' :: #:1:j → #:1:j → #:1:j
hole_#:1:j1_2 :: #:1:j
gen_#:1:j2_2 :: Nat → #:1:j

Lemmas:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)
opp(gen_#:1:j2_2(+(1, n6451040_2))) → *3_2, rt ∈ Ω(n64510402)
*'(gen_#:1:j2_2(n6454175_2), gen_#:1:j2_2(0)) → gen_#:1:j2_2(0), rt ∈ Ω(1 + n64541752)

Generator Equations:
gen_#:1:j2_2(0) ⇔ #
gen_#:1:j2_2(+(x, 1)) ⇔ 1(gen_#:1:j2_2(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)

### (20) Obligation:

TRS:
Rules:
0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

Types:
0 :: #:1:j → #:1:j
# :: #:1:j
+' :: #:1:j → #:1:j → #:1:j
1 :: #:1:j → #:1:j
j :: #:1:j → #:1:j
opp :: #:1:j → #:1:j
- :: #:1:j → #:1:j → #:1:j
*' :: #:1:j → #:1:j → #:1:j
hole_#:1:j1_2 :: #:1:j
gen_#:1:j2_2 :: Nat → #:1:j

Lemmas:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)
opp(gen_#:1:j2_2(+(1, n6451040_2))) → *3_2, rt ∈ Ω(n64510402)
*'(gen_#:1:j2_2(n6454175_2), gen_#:1:j2_2(0)) → gen_#:1:j2_2(0), rt ∈ Ω(1 + n64541752)

Generator Equations:
gen_#:1:j2_2(0) ⇔ #
gen_#:1:j2_2(+(x, 1)) ⇔ 1(gen_#:1:j2_2(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)

### (23) Obligation:

TRS:
Rules:
0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

Types:
0 :: #:1:j → #:1:j
# :: #:1:j
+' :: #:1:j → #:1:j → #:1:j
1 :: #:1:j → #:1:j
j :: #:1:j → #:1:j
opp :: #:1:j → #:1:j
- :: #:1:j → #:1:j → #:1:j
*' :: #:1:j → #:1:j → #:1:j
hole_#:1:j1_2 :: #:1:j
gen_#:1:j2_2 :: Nat → #:1:j

Lemmas:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)
opp(gen_#:1:j2_2(+(1, n6451040_2))) → *3_2, rt ∈ Ω(n64510402)

Generator Equations:
gen_#:1:j2_2(0) ⇔ #
gen_#:1:j2_2(+(x, 1)) ⇔ 1(gen_#:1:j2_2(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)

### (26) Obligation:

TRS:
Rules:
0(#) → #
+'(#, x) → x
+'(x, #) → x
+'(0(x), 0(y)) → 0(+'(x, y))
+'(0(x), 1(y)) → 1(+'(x, y))
+'(1(x), 0(y)) → 1(+'(x, y))
+'(0(x), j(y)) → j(+'(x, y))
+'(j(x), 0(y)) → j(+'(x, y))
+'(1(x), 1(y)) → j(+'(+'(x, y), 1(#)))
+'(j(x), j(y)) → 1(+'(+'(x, y), j(#)))
+'(1(x), j(y)) → 0(+'(x, y))
+'(j(x), 1(y)) → 0(+'(x, y))
+'(+'(x, y), z) → +'(x, +'(y, z))
opp(#) → #
opp(0(x)) → 0(opp(x))
opp(1(x)) → j(opp(x))
opp(j(x)) → 1(opp(x))
-(x, y) → +'(x, opp(y))
*'(#, x) → #
*'(0(x), y) → 0(*'(x, y))
*'(1(x), y) → +'(0(*'(x, y)), y)
*'(j(x), y) → -(0(*'(x, y)), y)
*'(*'(x, y), z) → *'(x, *'(y, z))

Types:
0 :: #:1:j → #:1:j
# :: #:1:j
+' :: #:1:j → #:1:j → #:1:j
1 :: #:1:j → #:1:j
j :: #:1:j → #:1:j
opp :: #:1:j → #:1:j
- :: #:1:j → #:1:j → #:1:j
*' :: #:1:j → #:1:j → #:1:j
hole_#:1:j1_2 :: #:1:j
gen_#:1:j2_2 :: Nat → #:1:j

Lemmas:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)

Generator Equations:
gen_#:1:j2_2(0) ⇔ #
gen_#:1:j2_2(+(x, 1)) ⇔ 1(gen_#:1:j2_2(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:1:j2_2(+(1, n4_2)), gen_#:1:j2_2(+(1, n4_2))) → *3_2, rt ∈ Ω(n42)