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Proof step 19 from proof gcdexh (previous step)

n < m, n * m0 + gcd(n, m - m / n * n) = (m - m / n * n) * n0 ∨ (m - m / n * n) * n0 + gcd(n, m - m / n * n) = n * m0, ∀ m0, n. m0 < m → n ≤ m0 → (∃ m1, n0. m0 * m1 + gcd(m0, n) = n * n0 ∨ n * n0 + gcd(m0, n) = m0 * m1) ⊦ m = n, n = 0, m * n0 + gcd(n, m - m / n * n) = n * (m0 + m / n * n0), n * (m0 + m / n * n0) + gcd(n, m - m / n * n) = m * n0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ m0 * (m + n) = m0 * m + m0 * n
Proof step 21:
n < m, n * m0 + gcd(n, m - m / n * n) = (m - m / n * n) * n0 ∨ (m - m / n * n) * n0 + gcd(n, m - m / n * n) = n * m0, ∀ m0, n. m0 < m → n ≤ m0 → (∃ m1, n0. m0 * m1 + gcd(m0, n) = n * n0 ∨ n * n0 + gcd(m0, n) = m0 * m1) ⊦ m = n, n = 0, m * n0 + gcd(n, m - m / n * n) = n * m0 + n * m / n * n0, (n * m0 + n * m / n * n0) + gcd(n, m - m / n * n) = m * n0

Rule argument

spec: nat-mult inst: name: dist seq: ⊦ m0 * (m + n) = m0 * m + m0 * n subst: [n, m, m0] → [m / n * n0, m0, n] flags: rewrite

Additional information

Used terms: m / n * n0, m0, n