The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0, E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)

The (relative) TRS S consists of the following rules:

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0, S(y)) → False
!EQ(S(x), 0) → False
!EQ(0, 0) → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

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:

getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)

The (relative) TRS S consists of the following rules:

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
via, !EQ, member, reach

They will be analysed ascendingly in the following order:
!EQ < via
via = reach
member < reach

(6) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

Generator Equations:
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))
gen_Cons:Nil6_0(0) ⇔ Nil
gen_Cons:Nil6_0(+(x, 1)) ⇔ Cons(E(0', 0'), gen_Cons:Nil6_0(x))

The following defined symbols remain to be analysed:
!EQ, via, member, reach

They will be analysed ascendingly in the following order:
!EQ < via
via = reach
member < reach

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

Proved the following rewrite lemma:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)

Induction Base:
!EQ(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) →_R^Ω(0)
False

Induction Step:
!EQ(gen_S:0'5_0(+(n8_0, 1)), gen_S:0'5_0(+(1, +(n8_0, 1)))) →_R^Ω(0)
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) →_IH
False

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n⁰).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))
gen_Cons:Nil6_0(0) ⇔ Nil
gen_Cons:Nil6_0(+(x, 1)) ⇔ Cons(E(0', 0'), gen_Cons:Nil6_0(x))

The following defined symbols remain to be analysed:
member, via, reach

They will be analysed ascendingly in the following order:
via = reach
member < reach

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))
gen_Cons:Nil6_0(0) ⇔ Nil
gen_Cons:Nil6_0(+(x, 1)) ⇔ Cons(E(0', 0'), gen_Cons:Nil6_0(x))

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

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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))
gen_Cons:Nil6_0(0) ⇔ Nil
gen_Cons:Nil6_0(+(x, 1)) ⇔ Cons(E(0', 0'), gen_Cons:Nil6_0(x))

The following defined symbols remain to be analysed:
via

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

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

Proved the following rewrite lemma:
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(n4239_0), gen_Cons:Nil6_0(d)) → gen_Cons:Nil6_0(0), rt ∈ Ω(1 + n4239₀)

Induction Base:
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(0), gen_Cons:Nil6_0(d)) →_R^Ω(1)
Nil

Induction Step:
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(+(n4239_0, 1)), gen_Cons:Nil6_0(d)) →_R^Ω(1)
via[Ite](!EQ(gen_S:0'5_0(1), 0'), gen_S:0'5_0(1), gen_S:0'5_0(b), Cons(E(0', 0'), gen_Cons:Nil6_0(n4239_0)), gen_Cons:Nil6_0(d)) →_R^Ω(0)
via[Ite](False, gen_S:0'5_0(1), gen_S:0'5_0(b), Cons(E(0', 0'), gen_Cons:Nil6_0(n4239_0)), gen_Cons:Nil6_0(d)) →_R^Ω(0)
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(n4239_0), gen_Cons:Nil6_0(d)) →_IH
gen_Cons:Nil6_0(0)

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

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(n4239_0), gen_Cons:Nil6_0(d)) → gen_Cons:Nil6_0(0), rt ∈ Ω(1 + n4239₀)

Generator Equations:
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))
gen_Cons:Nil6_0(0) ⇔ Nil
gen_Cons:Nil6_0(+(x, 1)) ⇔ Cons(E(0', 0'), gen_Cons:Nil6_0(x))

The following defined symbols remain to be analysed:
reach

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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(n4239_0), gen_Cons:Nil6_0(d)) → gen_Cons:Nil6_0(0), rt ∈ Ω(1 + n4239₀)

Generator Equations:
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))
gen_Cons:Nil6_0(0) ⇔ Nil
gen_Cons:Nil6_0(+(x, 1)) ⇔ Cons(E(0', 0'), gen_Cons:Nil6_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(n4239_0), gen_Cons:Nil6_0(d)) → gen_Cons:Nil6_0(0), rt ∈ Ω(1 + n4239₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(n4239_0), gen_Cons:Nil6_0(d)) → gen_Cons:Nil6_0(0), rt ∈ Ω(1 + n4239₀)

Generator Equations:
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))
gen_Cons:Nil6_0(0) ⇔ Nil
gen_Cons:Nil6_0(+(x, 1)) ⇔ Cons(E(0', 0'), gen_Cons:Nil6_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
via(gen_S:0'5_0(1), gen_S:0'5_0(b), gen_Cons:Nil6_0(n4239_0), gen_Cons:Nil6_0(d)) → gen_Cons:Nil6_0(0), rt ∈ Ω(1 + n4239₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
getNodeFromEdge(S(S(x')), E(x, y)) → y
via(u, v, Cons(E(x, y), xs), edges) → via[Ite](!EQ(u, x), u, v, Cons(E(x, y), xs), edges)
getNodeFromEdge(S(0'), E(x, y)) → x
member(x', Cons(x, xs)) → member[Ite](eqEdge(x', x), x', Cons(x, xs))
getNodeFromEdge(0', E(x, y)) → x
eqEdge(E(e11, e12), E(e21, e22)) → eqEdge[Ite](and(!EQ(e11, e21), !EQ(e12, e22)), e21, e22, e11, e12)
via(u, v, Nil, edges) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
member(x, Nil) → False
reach(u, v, edges) → reach[Ite](member(E(u, v), edges), u, v, edges)
goal(u, v, edges) → reach(u, v, edges)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
via[Ite](True, u, v, Cons(E(x, y), xs), edges) → via[Let](u, v, Cons(E(x, y), xs), edges, reach(y, v, edges))
via[Let](u, v, Cons(x, xs), edges, Nil) → via(u, v, xs, edges)
via[Let](u, v, Cons(x, xs), edges, Cons(x', xs')) → Cons(x, Cons(x', xs'))
via[Ite](False, u, v, Cons(x, xs), edges) → via(u, v, xs, edges)
member[Ite](False, x', Cons(x, xs)) → member(x', xs)
reach[Ite](False, u, v, edges) → via(u, v, edges, edges)
reach[Ite](True, u, v, edges) → Cons(E(u, v), Nil)
member[Ite](True, x, xs) → True
eqEdge[Ite](False, e21, e22, e11, e12) → False
eqEdge[Ite](True, e21, e22, e11, e12) → True

Types:
getNodeFromEdge :: S:0' → E → S:0'
S :: S:0' → S:0'
E :: S:0' → S:0' → E
via :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: E → Cons:Nil → Cons:Nil
via[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
0' :: S:0'
member :: E → Cons:Nil → True:False
member[Ite] :: True:False → E → Cons:Nil → True:False
eqEdge :: E → E → True:False
eqEdge[Ite] :: True:False → S:0' → S:0' → S:0' → S:0' → True:False
and :: True:False → True:False → True:False
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
reach :: S:0' → S:0' → Cons:Nil → Cons:Nil
reach[Ite] :: True:False → S:0' → S:0' → Cons:Nil → Cons:Nil
goal :: S:0' → S:0' → Cons:Nil → Cons:Nil
via[Let] :: S:0' → S:0' → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_S:0'1_0 :: S:0'
hole_E2_0 :: E
hole_Cons:Nil3_0 :: Cons:Nil
hole_True:False4_0 :: True:False
gen_S:0'5_0 :: Nat → S:0'
gen_Cons:Nil6_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))
gen_Cons:Nil6_0(0) ⇔ Nil
gen_Cons:Nil6_0(+(x, 1)) ⇔ Cons(E(0', 0'), gen_Cons:Nil6_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(1) was proven with the following lemma:
!EQ(gen_S:0'5_0(n8_0), gen_S:0'5_0(+(1, n8_0))) → False, rt ∈ Ω(0)

(26) BOUNDS(1, INF)