[Imprint][Data Privacy Statement][E-Mail]



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 21 from proof align-down-add-ls (previous step)

n0 % m + n1 < m, n0 ≤ (n0 + n1) / m * m ⊦ (n0 + n1) % m = 0, n0 % m = 0

User applied rule insert elim lemma

And closed the goal!

Rule argument

spec: nat-div name: div-elim seq: n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n subst: [m, n, m0, n0] → [n0, m, m0, n]

Additional information

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