### (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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
eq(s(x), s(y)) →+ eq(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:

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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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
union :: empty:edge → empty:edge → empty:edge
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
reach :: 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_1 :: true:false → 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_2 :: true:false → 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

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

Heuristically decided to analyse the following defined symbols:
eq, union, reach

They will be analysed ascendingly in the following order:
eq < reach
union < reach

### (8) Obligation:

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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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
union :: empty:edge → empty:edge → empty:edge
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
reach :: 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_1 :: true:false → 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_2 :: true:false → 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, union, reach

They will be analysed ascendingly in the following order:
eq < reach
union < reach

### (9) 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)) →RΩ(1)
true

Induction Step:
eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →RΩ(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 ircR ∈ Ω(n).

### (11) Obligation:

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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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
union :: empty:edge → empty:edge → empty:edge
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
reach :: 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_1 :: true:false → 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_2 :: true:false → 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:
union, reach

They will be analysed ascendingly in the following order:
union < reach

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

Proved the following rewrite lemma:
union(gen_empty:edge5_0(n540_0), gen_empty:edge5_0(b)) → gen_empty:edge5_0(+(n540_0, b)), rt ∈ Ω(1 + n5400)

Induction Base:
union(gen_empty:edge5_0(0), gen_empty:edge5_0(b)) →RΩ(1)
gen_empty:edge5_0(b)

Induction Step:
union(gen_empty:edge5_0(+(n540_0, 1)), gen_empty:edge5_0(b)) →RΩ(1)
edge(0', 0', union(gen_empty:edge5_0(n540_0), gen_empty:edge5_0(b))) →IH
edge(0', 0', gen_empty:edge5_0(+(b, c541_0)))

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

### (14) Obligation:

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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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
union :: empty:edge → empty:edge → empty:edge
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
reach :: 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_1 :: true:false → 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_2 :: true:false → 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)
union(gen_empty:edge5_0(n540_0), gen_empty:edge5_0(b)) → gen_empty:edge5_0(+(n540_0, b)), rt ∈ Ω(1 + n5400)

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

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

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

### (16) Obligation:

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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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
union :: empty:edge → empty:edge → empty:edge
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
reach :: 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_1 :: true:false → 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_2 :: true:false → 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)
union(gen_empty:edge5_0(n540_0), gen_empty:edge5_0(b)) → gen_empty:edge5_0(+(n540_0, b)), rt ∈ Ω(1 + n5400)

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.

### (17) 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)

### (19) Obligation:

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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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
union :: empty:edge → empty:edge → empty:edge
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
reach :: 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_1 :: true:false → 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_2 :: true:false → 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)
union(gen_empty:edge5_0(n540_0), gen_empty:edge5_0(b)) → gen_empty:edge5_0(+(n540_0, b)), rt ∈ Ω(1 + n5400)

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)

### (22) Obligation:

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
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
reach(x, y, empty, h) → false
reach(x, y, edge(u, v, i), h) → if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) → if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) → true
if_reach_2(false, x, y, edge(u, v, i), h) → or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) → reach(x, y, i, edge(u, v, h))

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
union :: empty:edge → empty:edge → empty:edge
empty :: empty:edge
edge :: 0':s → 0':s → empty:edge → empty:edge
reach :: 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_1 :: true:false → 0':s → 0':s → empty:edge → empty:edge → true:false
if_reach_2 :: true:false → 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.

### (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)