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Proof step 3 from proof div-elim (previous step)

⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n, n = 0

User applied rule insert lemma

And obtained the following new goals

Proof step 4:
⊦ n ≠ 0 → m = m / n * n + m % n

Rule argument

spec: <toplevel> name: Moddef seq: ⊦ n ≠ 0 → m = m / n * n + m % n subst: [n, m] → [n, m]

Additional information

Used simplifier rules: mod-ls: n ≠ 0 ⊦ m % n < n;%mult: n ≠ 0 ⊦ (n0 + m * n) / n = n0 / n + m;%zero-01: ⊦ m < n → m / n = 0;% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s) Used terms: n, m