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:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
+(x, +(y, z)) → +(+(x, y), z)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0) → true
ge(0, O(x)) → ge(0, x)
ge(0, I(x)) → false
Log'(0) → 0
Log'(I(x)) → +(Log'(x), I(0))
Log'(O(x)) → if(ge(x, I(0)), +(Log'(x), I(0)), 0)
Log(x) → -(Log'(x), I(0))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0)
Size(N(x, l, r)) → +(+(Size(l), Size(r)), I(1))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0), -(Size(l), Size(r))), ge(I(0), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

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

Heuristically decided to analyse the following defined symbols:
+', -, ge, Log', Min, Max, BS, Size, WB

They will be analysed ascendingly in the following order:
+' < Log'
+' < Size
- < WB
ge < Log'
ge < BS
ge < WB
Min < BS
Max < BS
Size < WB

(8) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
+', -, ge, Log', Min, Max, BS, Size, WB

They will be analysed ascendingly in the following order:
+' < Log'
+' < Size
- < WB
ge < Log'
ge < BS
ge < WB
Min < BS
Max < BS
Size < WB

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
-, ge, Log', Min, Max, BS, Size, WB

They will be analysed ascendingly in the following order:
- < WB
ge < Log'
ge < BS
ge < WB
Min < BS
Max < BS
Size < WB

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

Proved the following rewrite lemma:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)

Induction Base:
-(gen_0':I:1':true:false3_0(0), gen_0':I:1':true:false3_0(0)) →_R^Ω(1)
gen_0':I:1':true:false3_0(0)

Induction Step:
-(gen_0':I:1':true:false3_0(+(n365135_0, 1)), gen_0':I:1':true:false3_0(+(n365135_0, 1))) →_R^Ω(1)
O(-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0))) →_IH
O(gen_0':I:1':true:false3_0(0)) →_R^Ω(1)
0'

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
ge, Log', Min, Max, BS, Size, WB

They will be analysed ascendingly in the following order:
ge < Log'
ge < BS
ge < WB
Min < BS
Max < BS
Size < WB

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

Proved the following rewrite lemma:
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Induction Base:
ge(gen_0':I:1':true:false3_0(0), gen_0':I:1':true:false3_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':I:1':true:false3_0(+(n367257_0, 1)), gen_0':I:1':true:false3_0(+(n367257_0, 1))) →_R^Ω(1)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) →_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:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
Log', Min, Max, BS, Size, WB

They will be analysed ascendingly in the following order:
Min < BS
Max < BS
Size < WB

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

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

(18) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
Min, Max, BS, Size, WB

They will be analysed ascendingly in the following order:
Min < BS
Max < BS
Size < WB

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
Max, BS, Size, WB

They will be analysed ascendingly in the following order:
Max < BS
Size < WB

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
BS, Size, WB

They will be analysed ascendingly in the following order:
Size < WB

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(24) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
Size, WB

They will be analysed ascendingly in the following order:
Size < WB

(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(26) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

The following defined symbols remain to be analysed:
WB

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(28) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)

(30) BOUNDS(n^1, INF)

(31) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)
ge(gen_0':I:1':true:false3_0(n367257_0), gen_0':I:1':true:false3_0(n367257_0)) → true, rt ∈ Ω(1 + n367257₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)

(33) BOUNDS(n^1, INF)

(34) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
+'(x, +'(y, z)) → +'(+'(x, y), z)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
not(true) → false
not(false) → true
and(x, true) → x
and(x, false) → false
if(true, x, y) → x
if(false, x, y) → y
ge(O(x), O(y)) → ge(x, y)
ge(O(x), I(y)) → not(ge(y, x))
ge(I(x), O(y)) → ge(x, y)
ge(I(x), I(y)) → ge(x, y)
ge(x, 0') → true
ge(0', O(x)) → ge(0', x)
ge(0', I(x)) → false
Log'(0') → 0'
Log'(I(x)) → +'(Log'(x), I(0'))
Log'(O(x)) → if(ge(x, I(0')), +'(Log'(x), I(0')), 0')
Log(x) → -(Log'(x), I(0'))
Val(L(x)) → x
Val(N(x, l, r)) → x
Min(L(x)) → x
Min(N(x, l, r)) → Min(l)
Max(L(x)) → x
Max(N(x, l, r)) → Max(r)
BS(L(x)) → true
BS(N(x, l, r)) → and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))
Size(L(x)) → I(0')
Size(N(x, l, r)) → +'(+'(Size(l), Size(r)), I(1'))
WB(L(x)) → true
WB(N(x, l, r)) → and(if(ge(Size(l), Size(r)), ge(I(0'), -(Size(l), Size(r))), ge(I(0'), -(Size(r), Size(l)))), and(WB(l), WB(r)))

Types:
O :: 0':I:1':true:false → 0':I:1':true:false
0' :: 0':I:1':true:false
+' :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
I :: 0':I:1':true:false → 0':I:1':true:false
- :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
1' :: 0':I:1':true:false
not :: 0':I:1':true:false → 0':I:1':true:false
true :: 0':I:1':true:false
false :: 0':I:1':true:false
and :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
if :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
ge :: 0':I:1':true:false → 0':I:1':true:false → 0':I:1':true:false
Log' :: 0':I:1':true:false → 0':I:1':true:false
Log :: 0':I:1':true:false → 0':I:1':true:false
Val :: L:l:r:N → 0':I:1':true:false
L :: 0':I:1':true:false → L:l:r:N
N :: 0':I:1':true:false → L:l:r:N → L:l:r:N → L:l:r:N
l :: L:l:r:N
r :: L:l:r:N
Min :: L:l:r:N → 0':I:1':true:false
Max :: L:l:r:N → 0':I:1':true:false
BS :: L:l:r:N → 0':I:1':true:false
Size :: L:l:r:N → 0':I:1':true:false
WB :: L:l:r:N → 0':I:1':true:false
hole_0':I:1':true:false1_0 :: 0':I:1':true:false
hole_L:l:r:N2_0 :: L:l:r:N
gen_0':I:1':true:false3_0 :: Nat → 0':I:1':true:false
gen_L:l:r:N4_0 :: Nat → L:l:r:N

Lemmas:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)

Generator Equations:
gen_0':I:1':true:false3_0(0) ⇔ 0'
gen_0':I:1':true:false3_0(+(x, 1)) ⇔ I(gen_0':I:1':true:false3_0(x))
gen_L:l:r:N4_0(0) ⇔ L(0')
gen_L:l:r:N4_0(+(x, 1)) ⇔ N(0', L(0'), gen_L:l:r:N4_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
-(gen_0':I:1':true:false3_0(n365135_0), gen_0':I:1':true:false3_0(n365135_0)) → gen_0':I:1':true:false3_0(0), rt ∈ Ω(1 + n365135₀)

(36) BOUNDS(n^1, INF)