The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

table → gen(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))))))))))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

table → gen(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: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
entry/0
entry/1

(4) Obligation:

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

table → gen(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: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
table → gen(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

(7) 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

(8) Obligation:

Innermost TRS:
Rules:
table → gen(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

(9) 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)) →_R^Ω(1)
true

Induction Step:
le(gen_0':s6_0(+(n8_0, 1)), gen_0':s6_0(+(n8_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
table → gen(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

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

Proved the following rewrite lemma:
plus(gen_0':s6_0(n343_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n343_0, b)), rt ∈ Ω(1 + n343₀)

Induction Base:
plus(gen_0':s6_0(0), gen_0':s6_0(b)) →_R^Ω(1)
gen_0':s6_0(b)

Induction Step:
plus(gen_0':s6_0(+(n343_0, 1)), gen_0':s6_0(b)) →_R^Ω(1)
s(plus(gen_0':s6_0(n343_0), gen_0':s6_0(b))) →_IH
s(gen_0':s6_0(+(b, c344_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
table → gen(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(n343_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n343_0, b)), rt ∈ Ω(1 + n343₀)

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

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

Proved the following rewrite lemma:
times(gen_0':s6_0(n1210_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1210_0, b)), rt ∈ Ω(1 + b·n1210₀ + n1210₀)

Induction Base:
times(gen_0':s6_0(0), gen_0':s6_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_0':s6_0(+(n1210_0, 1)), gen_0':s6_0(b)) →_R^Ω(1)
plus(gen_0':s6_0(b), times(gen_0':s6_0(n1210_0), gen_0':s6_0(b))) →_IH
plus(gen_0':s6_0(b), gen_0':s6_0(*(c1211_0, b))) →_L^{Ω(1 + b)}
gen_0':s6_0(+(b, *(n1210_0, b)))

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
table → gen(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(n343_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n343_0, b)), rt ∈ Ω(1 + n343₀)
times(gen_0':s6_0(n1210_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1210_0, b)), rt ∈ Ω(1 + b·n1210₀ + n1210₀)

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

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

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

(19) Obligation:

Innermost TRS:
Rules:
table → gen(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(n343_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n343_0, b)), rt ∈ Ω(1 + n343₀)
times(gen_0':s6_0(n1210_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1210_0, b)), rt ∈ Ω(1 + b·n1210₀ + n1210₀)

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

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

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

(21) Obligation:

Innermost TRS:
Rules:
table → gen(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(n343_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n343_0, b)), rt ∈ Ω(1 + n343₀)
times(gen_0':s6_0(n1210_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1210_0, b)), rt ∈ Ω(1 + b·n1210₀ + n1210₀)

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.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s6_0(n1210_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1210_0, b)), rt ∈ Ω(1 + b·n1210₀ + n1210₀)

(23) BOUNDS(n^2, INF)

(24) Obligation:

Innermost TRS:
Rules:
table → gen(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(n343_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n343_0, b)), rt ∈ Ω(1 + n343₀)
times(gen_0':s6_0(n1210_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1210_0, b)), rt ∈ Ω(1 + b·n1210₀ + n1210₀)

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.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s6_0(n1210_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1210_0, b)), rt ∈ Ω(1 + b·n1210₀ + n1210₀)

(26) BOUNDS(n^2, INF)

(27) Obligation:

Innermost TRS:
Rules:
table → gen(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(n343_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n343_0, b)), rt ∈ Ω(1 + n343₀)

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.

(28) 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)

(29) BOUNDS(n^1, INF)

(30) Obligation:

Innermost TRS:
Rules:
table → gen(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.

(31) 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) BOUNDS(n^1, INF)