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Proof step 9 from proof square-ls-03 (previous step)

⊦ (m * (m + 1) + 2 * m0) / 2 < ((m + 1) * (m + 2)) / 2, ¬ ((m + 1) * (m + 2)) / 2 ≤ ((n + 1) * (n + 2)) / 2, n < m, m0 ≤ m ∧ m ≤ n → (m * (m + 1) + 2 * m0) / 2 < ((n + 1) * (n + 2)) / 2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ m % 2 = 0 ∧ n % 2 = 0 → (m / 2 < n / 2 ↔ m < n)
Proof step 11:
¬((m * (m + 1) + 2 * m0) % 2 = 0 ∧ ((m + 1) * (m + 2)) % 2 = 0) ⊦ (m * (m + 1) + 2 * m0) / 2 < ((m + 1) * (m + 2)) / 2, ¬ ((m + 1) * (m + 2)) / 2 ≤ ((n + 1) * (n + 2)) / 2, n < m, m0 ≤ m ∧ m ≤ n → (m * (m + 1) + 2 * m0) / 2 < ((n + 1) * (n + 2)) / 2
Proof step 13:
(m * (m + 1) + 2 * m0) % 2 = 0 ∧ ((m + 1) * (m + 2)) % 2 = 0 ⊦ m * (m + 1) + 2 * m0 < (m + 1) * (m + 2), ¬ ((m + 1) * (m + 2)) / 2 ≤ ((n + 1) * (n + 2)) / 2, n < m, m0 ≤ m ∧ m ≤ n → (m * (m + 1) + 2 * m0) / 2 < ((n + 1) * (n + 2)) / 2

Rule argument

spec: nat-div inst: name: ls-even seq: ⊦ m % 2 = 0 ∧ n % 2 = 0 → (m / 2 < n / 2 ↔ m < n) subst: [n, m] → [(m + 1) * (m + 2), m * (m + 1) + 2 * m0] flags: rewrite, current

Additional information

Used terms: (m + 1) * (m + 2), m * (m + 1) + 2 * m0