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Proof step 28 from proof gcdexh (previous step)

(m - m / n * n) * n0 + gcd(n, m - m / n * n) = n * m0, n < m, ∀ 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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%ls: ⊦ m0 * m < m0 * n ↔ m0 ≠ 0 ∧ m < n; from specification nat-mult%cond-01: n ≠ 0 ⊦ ¬ m < m / n * n; from specification nat-div%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%eq-02: n ≤ m ⊦ (m - n) + n0 = n1 ↔ m + n0 = n + n1; from specification nat%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2%cond: n ≠ 0 ⊦ m / n * n ≤ m; from specification nat-div Used terms: n0, m / n * n, m