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Proof step 13 from proof square-ls-03 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ m % 2 = 0 ∧ n % 2 = 0 → (m / 2 < n / 2 ↔ m < n)
Proof step 15:
((m + 1) * (m + 2)) / 2 ≤ ((n + 1) * (n + 2)) / 2, m ≤ n, m0 ≤ m ⊦ (m * (m + 1) + 2 * m0) / 2 < ((m + 1) * (m + 2)) / 2, n < m, (m * (m + 1) + 2 * m0) / 2 < ((n + 1) * (n + 2)) / 2, (m * (m + 1) + 2 * m0) % 2 = 0 ∧ ((m + 1) * (m + 2)) % 2 = 0
Proof step 17:
(m * (m + 1) + 2 * m0) % 2 = 0, ((m + 1) * (m + 2)) % 2 = 0, ((m + 1) * (m + 2)) / 2 ≤ ((n + 1) * (n + 2)) / 2, m ≤ n, m0 ≤ m ⊦ m * (m + 1) + 2 * m0 < (m + 1) * (m + 2), n < m, (m * (m + 1) + 2 * m0) / 2 < ((n + 1) * (n + 2)) / 2

Rule argument

spec: <toplevel> 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: <all paths>

Additional information

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