The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(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))
*(+(x, y), z) → +(*(x, z), *(y, z))
*(x, +(y, z)) → +(*(x, y), *(x, 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 [ ].

(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:

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))
*'(+'(x, y), z) → +'(*'(x, z), *'(y, z))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))

S is empty.
Rewrite Strategy: FULL

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

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))
*'(+'(x, y), z) → +'(*'(x, z), *'(y, z))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, 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))
*'(+'(x, y), z) → +'(*'(x, z), *'(y, z))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, 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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol +'.

(10) 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))
*'(+'(x, y), z) → +'(*'(x, z), *'(y, z))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, 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, *'

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

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

(12) 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))
*'(+'(x, y), z) → +'(*'(x, z), *'(y, z))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, 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:
*'

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
*'(gen_#:1:j2_2(n6452509_2), gen_#:1:j2_2(0)) → gen_#:1:j2_2(0), rt ∈ Ω(1 + n6452509₂)

Induction Base:
*'(gen_#:1:j2_2(0), gen_#:1:j2_2(0)) →_R^Ω(1)
#

Induction Step:
*'(gen_#:1:j2_2(+(n6452509_2, 1)), gen_#:1:j2_2(0)) →_R^Ω(1)
+'(0(*'(gen_#:1:j2_2(n6452509_2), gen_#:1:j2_2(0))), gen_#:1:j2_2(0)) →_IH
+'(0(gen_#:1:j2_2(0)), gen_#:1:j2_2(0)) →_R^Ω(1)
+'(#, gen_#:1:j2_2(0)) →_R^Ω(1)
gen_#:1:j2_2(0)

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

(14) Complex Obligation (BEST)

(15) 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))
*'(+'(x, y), z) → +'(*'(x, z), *'(y, z))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, 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(n6452509_2), gen_#:1:j2_2(0)) → gen_#:1:j2_2(0), rt ∈ Ω(1 + n6452509₂)

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.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_#:1:j2_2(n6452509_2), gen_#:1:j2_2(0)) → gen_#:1:j2_2(0), rt ∈ Ω(1 + n6452509₂)

(17) BOUNDS(n^1, INF)

(18) 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))
*'(+'(x, y), z) → +'(*'(x, z), *'(y, z))
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, 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(n6452509_2), gen_#:1:j2_2(0)) → gen_#:1:j2_2(0), rt ∈ Ω(1 + n6452509₂)

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.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_#:1:j2_2(n6452509_2), gen_#:1:j2_2(0)) → gen_#:1:j2_2(0), rt ∈ Ω(1 + n6452509₂)

(20) BOUNDS(n^1, INF)