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Proof step 4 from proof square-inject (previous step)

2 * ((m + n) * (m + n + 1)) / 2 + 2 * n = (((m0 + n0) * (m0 + n0 + 1)) / 2 + n0) * 2 ⊦ ((m + n) * (m + n + 1)) / 2 + n = ((m0 + n0) * (m0 + n0 + 1)) / 2 + n0 ↔ m = m0 ∧ n = n0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ n % m = 0 ∧ m ≠ 0 → m * n / m = n
Proof step 6:
((m + n) * (m + n + 1)) % 2 = 0 ∧ 2 ≠ 0, (m + n) * (m + n + 1) + 2 * n = (((m0 + n0) * (m0 + n0 + 1)) / 2 + n0) * 2 ⊦ ((m + n) * (m + n + 1)) / 2 + n = ((m0 + n0) * (m0 + n0 + 1)) / 2 + n0 ↔ m = m0 ∧ n = n0

Rule argument

spec: nat-div inst: name: mult-div seq: ⊦ n % m = 0 ∧ m ≠ 0 → m * n / m = n subst: [n, m] → [(m + n) * (m + n + 1), 2] flags: rewrite, current

Additional information

Used simplifier rules: square-even-02: ⊦ (m * m + m) % 2 = 0;%rsucc-02: ⊦ m * (n + 1) = m * n + m; from specification nat-mult%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%coef-04: ⊦ m * n + m0 * n = m1 * n ↔ ¬ ¬(m + m0 = m1 ∨ n = 0); from specification nat-mult Used terms: (m + n) * (m + n + 1), 2