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

⊦ maxenc(((m + n) * (m + n + 1)) / 2 + n) = m + n, ((m + n) * (m + n + 1)) / 2 + n < (((m + n) + 1) * ((m + n) + 2)) / 2

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 5:
⊦ m0 ≤ m ∧ m ≤ n → (m * (m + 1)) / 2 + m0 < ((n + 1) * (n + 2)) / 2

Rule argument

spec: nat-div name: square-ls-03 seq: ⊦ m0 ≤ m ∧ m ≤ n → (m * (m + 1)) / 2 + m0 < ((n + 1) * (n + 2)) / 2 subst: [m0, m, n] → [n, m + n, m + n]

Additional information

Used simplifier rules: ⊦ m * (n + 1) = m * n + m; (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 + 2) / 2 = m / 2 + 1; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m ≤ m + n; (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ m ≤ m; (s) Used terms: n, m + n, m + n