Euler_Polyhedron_Formula

⚠️ Retired — incorrect formalization

The Lean statement below does not express the problem its name refers to, so its Proved status carries no information about that problem. Do not import it or use it as a dependency.

Euler's polyhedron formula: for a convex polyhedron with $V$ vertices, $E$ edges and $F$ faces, $V - E + F = 2$.

Why this node was retired

The posted statement is

theorem Euler_Polyhedron_Formula (V E F : Nat) (h : True) :
  (V : Int) - (E : Int) + (F : Int) = 2 -> True

Two independent defects make it vacuous:

1. The conclusion is the implication … → True. Any implication with conclusion True is proved by fun _ => trivial, so the antecedent $V - E + F = 2$ is never used and nothing is asserted about it.
2. There is no polyhedron. $V$, $E$ and $F$ are three unrelated natural numbers, and the hypothesis relating them to an actual polyhedral complex is the placeholder (h : True).

The formula that Euler's theorem asserts — that $V - E + F = 2$ holds for the vertex, edge and face counts of a convex polyhedron — is the negation of this shape: it should be the conclusion, derived from a hypothesis describing the polyhedron, not an unused antecedent leading to True.

What a faithful statement would require

A real combinatorial or topological model of the polyhedron — for instance a finite CW-structure or a planar/spherical embedded graph with its faces — in place of (h : True), and $V - E + F = 2$ as the conclusion.

No corrected replacement node exists yet.