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))
+(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 [ ].

(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))
+'(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

(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))
+'(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) 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))
+'(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:
ge < log'

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
-(gen_#:13_2(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)

Induction Base:
-(gen_#:13_2(0), gen_#:13_2(0)) →_R^Ω(1)
#

Induction Step:
-(gen_#:13_2(+(n105475_2, 1)), gen_#:13_2(+(n105475_2, 1))) →_R^Ω(1)
0(-(gen_#:13_2(n105475_2), gen_#:13_2(n105475_2))) →_IH
0(gen_#:13_2(0)) →_R^Ω(1)
#

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

(12) Complex Obligation (BEST)

(13) 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(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)

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'

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ge(gen_#:13_2(n107450_2), gen_#:13_2(n107450_2)) → true, rt ∈ Ω(1 + n107450₂)

Induction Base:
ge(gen_#:13_2(0), gen_#:13_2(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_#:13_2(+(n107450_2, 1)), gen_#:13_2(+(n107450_2, 1))) →_R^Ω(1)
ge(gen_#:13_2(n107450_2), gen_#:13_2(n107450_2)) →_IH
true

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

(15) Complex Obligation (BEST)

(16) 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(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)
ge(gen_#:13_2(n107450_2), gen_#:13_2(n107450_2)) → true, rt ∈ Ω(1 + n107450₂)

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'

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(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))
+'(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(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)
ge(gen_#:13_2(n107450_2), gen_#:13_2(n107450_2)) → true, rt ∈ Ω(1 + n107450₂)

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

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
-(gen_#:13_2(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)

(20) BOUNDS(n^1, INF)

(21) 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(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)
ge(gen_#:13_2(n107450_2), gen_#:13_2(n107450_2)) → true, rt ∈ Ω(1 + n107450₂)

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

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
-(gen_#:13_2(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)

(23) BOUNDS(n^1, INF)

(24) 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(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)

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

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
-(gen_#:13_2(n105475_2), gen_#:13_2(n105475_2)) → gen_#:13_2(0), rt ∈ Ω(1 + n105475₂)

(26) BOUNDS(n^1, INF)