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Proof step 26 from proof gcdexh (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
⊦ (m + n0) + n1 = m + n0 + n1

Rule argument

spec: nat-basic2 inst: name: a seq: ⊦ (m + n0) + n1 = m + n0 + n1 subst: [n1, n0, m] → [n * m / n * n0, gcd(n, m - m / n * n), n * m0] flags: rewrite

Additional information

Used simplifier rules: coef-04: ⊦ m * n + m0 * n = m1 * n ↔ ¬ ¬(m + m0 = m1 ∨ n = 0); from specification nat-mult%cond: n ≠ 0 ⊦ m / n * n ≤ m; 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%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2 Used terms: n * m / n * n0, gcd(n, m - m / n * n), n * m0