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Proof step 14 from proof alignup-le-is-aligned (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ n ≠ 0 → m % n = m - m / n * n
Proof step 16:
m - m / n * n = 0, m0 = n0 + m0 / n * n, m0 < n * m0 / n + n, m < n * m0 / n + n, n0 < n, m0 ≤ m ⊦ n = 0, n0 = 0

Rule argument

spec: nat-div name: mod-rew seq: ⊦ n ≠ 0 → m % n = m - m / n * n subst: [n, m] → [n, m] flags: <all paths>

Additional information

Used simplifier rules: ⊦ 0 * n = 0; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ¬ m < 0; (s) Used terms: n, m