### (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))
+(1(x), 1(y)) → 0(+(+(x, y), 1(#)))
+(+(x, y), z) → +(x, +(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +(log'(x), 1(#)), #)

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)) →+ 0(+(+(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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

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

Heuristically decided to analyse the following defined symbols:
+', -, ge, log'

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

### (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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

The following defined symbols remain to be analysed:
+', -, ge, log'

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

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

Proved the following rewrite lemma:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)

Induction Base:
+'(gen_#:13_2(0), gen_#:13_2(0))

Induction Step:
+'(gen_#:13_2(+(n5_2, 1)), gen_#:13_2(+(n5_2, 1))) →RΩ(1)
0(+'(+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)), 1(#))) →IH
0(+'(*4_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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Lemmas:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

The following defined symbols remain to be analysed:
-, ge, log'

They will be analysed ascendingly in the following order:
ge < log'

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

Proved the following rewrite lemma:
-(gen_#:13_2(n105625_2), gen_#:13_2(n105625_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1056252)

Induction Base:
-(gen_#:13_2(0), gen_#:13_2(0)) →RΩ(1)
#

Induction Step:
-(gen_#:13_2(+(n105625_2, 1)), gen_#:13_2(+(n105625_2, 1))) →RΩ(1)
0(-(gen_#:13_2(n105625_2), gen_#:13_2(n105625_2))) →IH
0(gen_#:13_2(0)) →RΩ(1)
#

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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Lemmas:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)
-(gen_#:13_2(n105625_2), gen_#:13_2(n105625_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1056252)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

The following defined symbols remain to be analysed:
ge, log'

They will be analysed ascendingly in the following order:
ge < log'

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

Proved the following rewrite lemma:
ge(gen_#:13_2(n107664_2), gen_#:13_2(n107664_2)) → true, rt ∈ Ω(1 + n1076642)

Induction Base:
ge(gen_#:13_2(0), gen_#:13_2(0)) →RΩ(1)
true

Induction Step:
ge(gen_#:13_2(+(n107664_2, 1)), gen_#:13_2(+(n107664_2, 1))) →RΩ(1)
ge(gen_#:13_2(n107664_2), gen_#:13_2(n107664_2)) →IH
true

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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Lemmas:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)
-(gen_#:13_2(n105625_2), gen_#:13_2(n105625_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1056252)
ge(gen_#:13_2(n107664_2), gen_#:13_2(n107664_2)) → true, rt ∈ Ω(1 + n1076642)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

The following defined symbols remain to be analysed:
log'

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

Proved the following rewrite lemma:
log'(gen_#:13_2(+(1, n111281_2))) → *4_2, rt ∈ Ω(n1112812)

Induction Base:
log'(gen_#:13_2(+(1, 0)))

Induction Step:
log'(gen_#:13_2(+(1, +(n111281_2, 1)))) →RΩ(1)
+'(log'(gen_#:13_2(+(1, n111281_2))), 1(#)) →IH
+'(*4_2, 1(#))

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

### (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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Lemmas:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)
-(gen_#:13_2(n105625_2), gen_#:13_2(n105625_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1056252)
ge(gen_#:13_2(n107664_2), gen_#:13_2(n107664_2)) → true, rt ∈ Ω(1 + n1076642)
log'(gen_#:13_2(+(1, n111281_2))) → *4_2, rt ∈ Ω(n1112812)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)

### (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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Lemmas:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)
-(gen_#:13_2(n105625_2), gen_#:13_2(n105625_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1056252)
ge(gen_#:13_2(n107664_2), gen_#:13_2(n107664_2)) → true, rt ∈ Ω(1 + n1076642)
log'(gen_#:13_2(+(1, n111281_2))) → *4_2, rt ∈ Ω(n1112812)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)

### (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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Lemmas:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)
-(gen_#:13_2(n105625_2), gen_#:13_2(n105625_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1056252)
ge(gen_#:13_2(n107664_2), gen_#:13_2(n107664_2)) → true, rt ∈ Ω(1 + n1076642)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)

### (29) 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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Lemmas:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)
-(gen_#:13_2(n105625_2), gen_#:13_2(n105625_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n1056252)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)

### (32) 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))
+'(1(x), 1(y)) → 0(+'(+'(x, y), 1(#)))
+'(+'(x, y), z) → +'(x, +'(y, z))
-(#, x) → #
-(x, #) → x
-(0(x), 0(y)) → 0(-(x, y))
-(0(x), 1(y)) → 1(-(-(x, y), 1(#)))
-(1(x), 0(y)) → 1(-(x, y))
-(1(x), 1(y)) → 0(-(x, y))
not(true) → false
not(false) → true
if(true, x, y) → x
if(false, x, y) → y
ge(0(x), 0(y)) → ge(x, y)
ge(0(x), 1(y)) → not(ge(y, x))
ge(1(x), 0(y)) → ge(x, y)
ge(1(x), 1(y)) → ge(x, y)
ge(x, #) → true
ge(#, 0(x)) → ge(#, x)
ge(#, 1(x)) → false
log(x) → -(log'(x), 1(#))
log'(#) → #
log'(1(x)) → +'(log'(x), 1(#))
log'(0(x)) → if(ge(x, 1(#)), +'(log'(x), 1(#)), #)

Types:
0 :: #:1 → #:1
# :: #:1
+' :: #:1 → #:1 → #:1
1 :: #:1 → #:1
- :: #:1 → #:1 → #:1
not :: true:false → true:false
true :: true:false
false :: true:false
if :: true:false → #:1 → #:1 → #:1
ge :: #:1 → #:1 → true:false
log :: #:1 → #:1
log' :: #:1 → #:1
hole_#:11_2 :: #:1
hole_true:false2_2 :: true:false
gen_#:13_2 :: Nat → #:1

Lemmas:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)

Generator Equations:
gen_#:13_2(0) ⇔ #
gen_#:13_2(+(x, 1)) ⇔ 1(gen_#:13_2(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) → *4_2, rt ∈ Ω(n52)