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:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
inc(0) → 0
inc(s(x)) → s(inc(x))
minus(0, y) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0)
log2(x, y) → if(le(x, 0), le(x, s(0)), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0))), y)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

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

Heuristically decided to analyse the following defined symbols:
le, inc, minus, quot, log2

They will be analysed ascendingly in the following order:
le < log2
inc < log2
minus < quot
quot < log2

(8) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

The following defined symbols remain to be analysed:
le, inc, minus, quot, log2

They will be analysed ascendingly in the following order:
le < log2
inc < log2
minus < quot
quot < log2

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

Proved the following rewrite lemma:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
le(gen_0':s:log_undefined3_0(0), gen_0':s:log_undefined3_0(0)) →_R^Ω(1)
true

Induction Step:
le(gen_0':s:log_undefined3_0(+(n5_0, 1)), gen_0':s:log_undefined3_0(+(n5_0, 1))) →_R^Ω(1)
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) →_IH
true

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

The following defined symbols remain to be analysed:
inc, minus, quot, log2

They will be analysed ascendingly in the following order:
inc < log2
minus < quot
quot < log2

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

Proved the following rewrite lemma:
inc(gen_0':s:log_undefined3_0(n312_0)) → gen_0':s:log_undefined3_0(n312_0), rt ∈ Ω(1 + n312₀)

Induction Base:
inc(gen_0':s:log_undefined3_0(0)) →_R^Ω(1)
0'

Induction Step:
inc(gen_0':s:log_undefined3_0(+(n312_0, 1))) →_R^Ω(1)
s(inc(gen_0':s:log_undefined3_0(n312_0))) →_IH
s(gen_0':s:log_undefined3_0(c313_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
inc(gen_0':s:log_undefined3_0(n312_0)) → gen_0':s:log_undefined3_0(n312_0), rt ∈ Ω(1 + n312₀)

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

The following defined symbols remain to be analysed:
minus, quot, log2

They will be analysed ascendingly in the following order:
minus < quot
quot < log2

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

Proved the following rewrite lemma:
minus(gen_0':s:log_undefined3_0(n526_0), gen_0':s:log_undefined3_0(n526_0)) → gen_0':s:log_undefined3_0(0), rt ∈ Ω(1 + n526₀)

Induction Base:
minus(gen_0':s:log_undefined3_0(0), gen_0':s:log_undefined3_0(0)) →_R^Ω(1)
0'

Induction Step:
minus(gen_0':s:log_undefined3_0(+(n526_0, 1)), gen_0':s:log_undefined3_0(+(n526_0, 1))) →_R^Ω(1)
minus(gen_0':s:log_undefined3_0(n526_0), gen_0':s:log_undefined3_0(n526_0)) →_IH
gen_0':s:log_undefined3_0(0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
inc(gen_0':s:log_undefined3_0(n312_0)) → gen_0':s:log_undefined3_0(n312_0), rt ∈ Ω(1 + n312₀)
minus(gen_0':s:log_undefined3_0(n526_0), gen_0':s:log_undefined3_0(n526_0)) → gen_0':s:log_undefined3_0(0), rt ∈ Ω(1 + n526₀)

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

The following defined symbols remain to be analysed:
quot, log2

They will be analysed ascendingly in the following order:
quot < log2

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
inc(gen_0':s:log_undefined3_0(n312_0)) → gen_0':s:log_undefined3_0(n312_0), rt ∈ Ω(1 + n312₀)
minus(gen_0':s:log_undefined3_0(n526_0), gen_0':s:log_undefined3_0(n526_0)) → gen_0':s:log_undefined3_0(0), rt ∈ Ω(1 + n526₀)

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

The following defined symbols remain to be analysed:
log2

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol log2.

(21) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
inc(gen_0':s:log_undefined3_0(n312_0)) → gen_0':s:log_undefined3_0(n312_0), rt ∈ Ω(1 + n312₀)
minus(gen_0':s:log_undefined3_0(n526_0), gen_0':s:log_undefined3_0(n526_0)) → gen_0':s:log_undefined3_0(0), rt ∈ Ω(1 + n526₀)

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
inc(gen_0':s:log_undefined3_0(n312_0)) → gen_0':s:log_undefined3_0(n312_0), rt ∈ Ω(1 + n312₀)
minus(gen_0':s:log_undefined3_0(n526_0), gen_0':s:log_undefined3_0(n526_0)) → gen_0':s:log_undefined3_0(0), rt ∈ Ω(1 + n526₀)

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(26) BOUNDS(n^1, INF)

(27) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
inc(gen_0':s:log_undefined3_0(n312_0)) → gen_0':s:log_undefined3_0(n312_0), rt ∈ Ω(1 + n312₀)

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(29) BOUNDS(n^1, INF)

(30) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(0') → 0'
inc(s(x)) → s(inc(x))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined → 0':s:log_undefined → true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined → 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined → 0':s:log_undefined
minus :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
quot :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log :: 0':s:log_undefined → 0':s:log_undefined
log2 :: 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
if :: true:false → true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false → 0':s:log_undefined → 0':s:log_undefined → 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat → 0':s:log_undefined

Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:log_undefined3_0(0) ⇔ 0'
gen_0':s:log_undefined3_0(+(x, 1)) ⇔ s(gen_0':s:log_undefined3_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(32) BOUNDS(n^1, INF)