### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

tablegen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10s(s(s(s(s(s(s(s(s(s(0))))))))))

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 [ ].

### (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:

tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
entry/0
entry/1

### (6) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
gen, le, if2, times, plus

They will be analysed ascendingly in the following order:
le < gen
gen = if2
le < if2
times < if2
plus < times

### (10) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

The following defined symbols remain to be analysed:
le, gen, if2, times, plus

They will be analysed ascendingly in the following order:
le < gen
gen = if2
le < if2
times < if2
plus < times

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

Proved the following rewrite lemma:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)

Induction Base:
le(gen_0':s6_0(0), gen_0':s6_0(0)) →RΩ(1)
true

Induction Step:
le(gen_0':s6_0(+(n8_0, 1)), gen_0':s6_0(+(n8_0, 1))) →RΩ(1)
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) →IH
true

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

### (13) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

The following defined symbols remain to be analysed:
plus, gen, if2, times

They will be analysed ascendingly in the following order:
gen = if2
times < if2
plus < times

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

Proved the following rewrite lemma:
plus(gen_0':s6_0(n313_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n313_0, b)), rt ∈ Ω(1 + n3130)

Induction Base:
plus(gen_0':s6_0(0), gen_0':s6_0(b)) →RΩ(1)
gen_0':s6_0(b)

Induction Step:
plus(gen_0':s6_0(+(n313_0, 1)), gen_0':s6_0(b)) →RΩ(1)
s(plus(gen_0':s6_0(n313_0), gen_0':s6_0(b))) →IH
s(gen_0':s6_0(+(b, c314_0)))

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

### (16) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)
plus(gen_0':s6_0(n313_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n313_0, b)), rt ∈ Ω(1 + n3130)

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

The following defined symbols remain to be analysed:
times, gen, if2

They will be analysed ascendingly in the following order:
gen = if2
times < if2

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

Proved the following rewrite lemma:
times(gen_0':s6_0(n1130_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1130_0, b)), rt ∈ Ω(1 + b·n11300 + n11300)

Induction Base:
times(gen_0':s6_0(0), gen_0':s6_0(b)) →RΩ(1)
0'

Induction Step:
times(gen_0':s6_0(+(n1130_0, 1)), gen_0':s6_0(b)) →RΩ(1)
plus(gen_0':s6_0(b), times(gen_0':s6_0(n1130_0), gen_0':s6_0(b))) →IH
plus(gen_0':s6_0(b), gen_0':s6_0(*(c1131_0, b))) →LΩ(1 + b)
gen_0':s6_0(+(b, *(n1130_0, b)))

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (19) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)
plus(gen_0':s6_0(n313_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n313_0, b)), rt ∈ Ω(1 + n3130)
times(gen_0':s6_0(n1130_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1130_0, b)), rt ∈ Ω(1 + b·n11300 + n11300)

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

The following defined symbols remain to be analysed:
if2, gen

They will be analysed ascendingly in the following order:
gen = if2

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

Could not prove a rewrite lemma for the defined symbol if2.

### (21) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)
plus(gen_0':s6_0(n313_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n313_0, b)), rt ∈ Ω(1 + n3130)
times(gen_0':s6_0(n1130_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1130_0, b)), rt ∈ Ω(1 + b·n11300 + n11300)

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

The following defined symbols remain to be analysed:
gen

They will be analysed ascendingly in the following order:
gen = if2

### (22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (23) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)
plus(gen_0':s6_0(n313_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n313_0, b)), rt ∈ Ω(1 + n3130)
times(gen_0':s6_0(n1130_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1130_0, b)), rt ∈ Ω(1 + b·n11300 + n11300)

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s6_0(n1130_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1130_0, b)), rt ∈ Ω(1 + b·n11300 + n11300)

### (26) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)
plus(gen_0':s6_0(n313_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n313_0, b)), rt ∈ Ω(1 + n3130)
times(gen_0':s6_0(n1130_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1130_0, b)), rt ∈ Ω(1 + b·n11300 + n11300)

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s6_0(n1130_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1130_0, b)), rt ∈ Ω(1 + b·n11300 + n11300)

### (29) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)
plus(gen_0':s6_0(n313_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n313_0, b)), rt ∈ Ω(1 + n3130)

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)

### (32) Obligation:

TRS:
Rules:
tablegen(s(0'))
gen(x) → if1(le(x, 10'), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10'), x, y)
if3(true, x, y) → cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
10's(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s → nil:cons
s :: 0':s → 0':s
0' :: 0':s
if1 :: false:true → 0':s → nil:cons
le :: 0':s → 0':s → false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s → 0':s → nil:cons
if3 :: false:true → 0':s → 0':s → nil:cons
cons :: entry → nil:cons → nil:cons
entry :: 0':s → entry
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat → nil:cons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)

Generator Equations:
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) → true, rt ∈ Ω(1 + n80)