The Runtime Complexity (innermost) 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:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0
size(edge(x, y, i)) → s(size(i))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0, i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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:

eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
eq, size, le, reach, if2

They will be analysed ascendingly in the following order:
eq < reach
eq < if2
size < reach
le < reach
reach = if2

(6) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

The following defined symbols remain to be analysed:
eq, size, le, reach, if2

They will be analysed ascendingly in the following order:
eq < reach
eq < if2
size < reach
le < reach
reach = if2

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →_R^Ω(1)
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →_IH
true

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

The following defined symbols remain to be analysed:
size, le, reach, if2

They will be analysed ascendingly in the following order:
size < reach
le < reach
reach = if2

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
size(gen_empty:edge5_0(n588_0)) → gen_0':s4_0(n588_0), rt ∈ Ω(1 + n588₀)

Induction Base:
size(gen_empty:edge5_0(0)) →_R^Ω(1)
0'

Induction Step:
size(gen_empty:edge5_0(+(n588_0, 1))) →_R^Ω(1)
s(size(gen_empty:edge5_0(n588_0))) →_IH
s(gen_0':s4_0(c589_0))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
size(gen_empty:edge5_0(n588_0)) → gen_0':s4_0(n588_0), rt ∈ Ω(1 + n588₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

The following defined symbols remain to be analysed:
le, reach, if2

They will be analysed ascendingly in the following order:
le < reach
reach = if2

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
le(gen_0':s4_0(n864_0), gen_0':s4_0(n864_0)) → true, rt ∈ Ω(1 + n864₀)

Induction Base:
le(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
le(gen_0':s4_0(+(n864_0, 1)), gen_0':s4_0(+(n864_0, 1))) →_R^Ω(1)
le(gen_0':s4_0(n864_0), gen_0':s4_0(n864_0)) →_IH
true

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
size(gen_empty:edge5_0(n588_0)) → gen_0':s4_0(n588_0), rt ∈ Ω(1 + n588₀)
le(gen_0':s4_0(n864_0), gen_0':s4_0(n864_0)) → true, rt ∈ Ω(1 + n864₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

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

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

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
size(gen_empty:edge5_0(n588_0)) → gen_0':s4_0(n588_0), rt ∈ Ω(1 + n588₀)
le(gen_0':s4_0(n864_0), gen_0':s4_0(n864_0)) → true, rt ∈ Ω(1 + n864₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

The following defined symbols remain to be analysed:
reach

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

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

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

(19) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
size(gen_empty:edge5_0(n588_0)) → gen_0':s4_0(n588_0), rt ∈ Ω(1 + n588₀)
le(gen_0':s4_0(n864_0), gen_0':s4_0(n864_0)) → true, rt ∈ Ω(1 + n864₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
size(gen_empty:edge5_0(n588_0)) → gen_0':s4_0(n588_0), rt ∈ Ω(1 + n588₀)
le(gen_0':s4_0(n864_0), gen_0':s4_0(n864_0)) → true, rt ∈ Ω(1 + n864₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(24) BOUNDS(n^1, INF)

(25) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
size(gen_empty:edge5_0(n588_0)) → gen_0':s4_0(n588_0), rt ∈ Ω(1 + n588₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
or(true, y) → true
or(false, y) → y
and(true, y) → y
and(false, y) → false
size(empty) → 0'
size(edge(x, y, i)) → s(size(i))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
reachable(x, y, i) → reach(x, y, 0', i, i)
reach(x, y, c, i, j) → if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) → true
if1(false, x, y, c, i, j) → if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) → false
if2(true, x, y, c, empty, j) → false
if2(true, x, y, c, edge(u, v, i), j) → or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
or :: true:false → true:false → true:false
and :: true:false → true:false → true:false
size :: empty:edge → 0':s
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
le :: 0':s → 0':s → true:false
reachable :: 0':s → 0':s → empty:edge → true:false
reach :: 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if1 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
if2 :: true:false → 0':s → 0':s → 0':s → empty:edge → empty:edge → true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat → 0':s
gen_empty:edge5_0 :: Nat → empty:edge

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_empty:edge5_0(0) ⇔ empty
gen_empty:edge5_0(+(x, 1)) ⇔ edge(0', 0', gen_empty:edge5_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(30) BOUNDS(n^1, INF)