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Proof step 8 from proof square-interval (previous step)

2 * (j * (j + 1)) / 2 + 2 * j0 < 2 * ((i + 1) * (i + 2)) / 2, 0 ≤ j0, 0 ≤ j, 0 ≤ i, j0 ≤ j, (j * (j + 1)) / 2 + j0 < ((i + 1) * (i + 2)) / 2, (i * (i + 1)) / 2 ≤ (j * (j + 1)) / 2 + j0 ⊦ i = j, i * (i + 1) ≤ j * (j + 1) + 2 * j0 ∧ j * (j + 1) + 2 * j0 < (i + 1) * (i + 2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ i % 2 ≠ 1 ∧ 0 < i → i / 2 * 2 = i
Proof step 10:
((i + 1) * (i + 2)) % 2 ≠ 1 ∧ 0 < (i + 1) * (i + 2), 2 * (j * (j + 1)) / 2 + 2 * j0 < (i + 1) * (i + 2), 0 ≤ j0, 0 ≤ j, 0 ≤ i, j0 ≤ j, (j * (j + 1)) / 2 + j0 < ((i + 1) * (i + 2)) / 2, (i * (i + 1)) / 2 ≤ (j * (j + 1)) / 2 + j0 ⊦ i = j, i * (i + 1) ≤ j * (j + 1) + 2 * j0 ∧ j * (j + 1) + 2 * j0 < (i + 1) * (i + 2)

Rule argument

spec: int-div inst: name: div2zero seq: ⊦ i % 2 ≠ 1 ∧ 0 < i → i / 2 * 2 = i subst: [i] → [(i + 1) * (i + 2)] flags: rewrite, current

Additional information

Used simplifier rules: square-even-02: ⊦ ((i + 1) * (i + 2)) % 2 = 0;%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%square-zero-01: ⊦ 0 ≤ i → ¬ (i + 1) * (i + 2) ≤ 0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: (i + 1) * (i + 2)