Lean Proofs

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# Natural Numbers

Lemma: Let $n \in \mathbb{N}$. Then $0 + n = n$.

Lemma: Let $n, m \in \mathbb{N}$. Then $S(n) + m = S(n + m)$.

## Commutativity

Lemma: Let $n, m \in \mathbb{N}$. Then $n + m = m + n$.

# Subtraction

Truncated subtraction: if $n < m$ we assume $n = m = 0$.

# Multiplication

Lemma: One is the identity element for multiplication

# Divisibility

Lemma: One divides every natural number