Fundamental_Principle_of_Counting

⚠️ Retired — incorrect formalization

The Lean statement below asserts nothing: its conclusion is the proposition True, which is closed by trivial in any context. Its proof status therefore records no mathematical content and says nothing about Fundamental_Principle_of_Counting. Do not import it or use it as a dependency.

Let $A$ be a finite set. Let $\sequence {B_n}$ be a sequence of distinct finite subsets of $A$ which form a partition of $A$. Let $p_k = \size {B_k}$ for each $k \in \closedint 1 n$. Then $\displaystyle \size A = \sum_{k \mathop = 1}^n p_k$ That is, the sum of the numbers of elements in the subsets of a partition of a set is equal to the total number of elements in the set.

Why this node was retired

The posted statement is

theorem Fundamental_Principle_of_Counting (A : Finset ℕ) (B : ℕ → Finset ℕ) (hB : ∀ i j, i ≠ j → Disjoint (B i) (B j)) (hAB : ∀ x, x ∈ A ↔ ∃ i, x ∈ B i) : True := by sorry

and its goal is True. Since True holds unconditionally, the hypotheses are never used and no relationship among them is asserted — the statement can be closed without doing any part of the work its name refers to. Concretely, this node is a title attached to an empty proposition: it occupies the name of a real result while proving none of it, and anything that imports it gains nothing.

What a faithful statement would require

The conclusion must be the mathematical assertion that the name refers to, expressed in terms of the objects and hypotheses in the signature, in place of True.

No corrected replacement node exists yet.