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Proof step 21 from proof pot-square-est-02 (previous step)

i1 * (i1 + 1) * 4 + 1 * (i1 + 1) * 4 < i1 * i1 + 4 * i1 + 4, i1 * (i1 + 1) * 4 + 1 * (i1 + 1) * 4 < 4 ^ i1 * 4, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, 0 ≤ i1, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) ∧ 0 < i0 ∧ 1 < i0 → false ⊦ i1 * (i1 + 1) * 4 + 1 * (i1 + 1) * 4 = 4 * i1 * i1 + 8 * i1 + 4

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 23:
(i1 * 4 * i1 + 1 * 4 * i1) + 1 * (i1 + 1) * 4 < i1 * i1 + 4 * i1 + 4, (i1 * 4 * i1 + 1 * 4 * i1) + 1 * (i1 + 1) * 4 < 4 ^ i1 * 4, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, 0 ≤ i1, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) ∧ 0 < i0 ∧ 1 < i0 → false ⊦ (i1 * 4 * i1 + 1 * 4 * i1) + 1 * (i1 + 1) * 4 = 4 * i1 * i1 + 8 * i1 + 4

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [4 * i1, 1, i1] flags: rewrite

Additional information

Used simplifier rules: associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: 4 * i1, 1, i1