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Proof step 21 from proof gcd-mod-one (previous step)

n0 ≤ n0 * m, m0 * m = n * n0 - 1, m0 ≤ n * n0, gcd(m, n) = 1, m * m0 + 1 = n * n0, (n * n0) * m - (n * n0 - 1) = m1 * n + n1, m0 ≤ n * n0, n1 < n ⊦ ((n0 * m) * n - n0 * n) + 1 = (n * n0) * m - (n * n0 - 1), n1 = 1, n = 1, m = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: nat inst: name: sub-succ seq: ⊦ m0 ≤ n ∧ n ≤ m → m - (n - m0) = (m - n) + m0 subst: [m0, n, m] → [1, n * n0, (n * n0) * m] flags: rewrite

Additional information

Used simplifier rules: le-02: ⊦ m0 ≠ 0 → m ≤ m * m0; from specification nat-mult%lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%one-01: ⊦ m < 1 ↔ m = 0; from specification nat-basic1%mult-base: ⊦ m * 0 = 0; from specification nat-mult%z-01: ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: 1, n * n0, (n * n0) * m