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Proof step 22 from proof square-interval (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: div2zero seq: ⊦ i % 2 ≠ 1 ∧ 0 < i → i / 2 * 2 = i subst: [i] → [j * (j + 1)] flags: <all paths>

Additional information

Used simplifier rules: square-even: ⊦ 0 ≤ i → (i * (i + 1)) % 2 = 0;%square-even-02: ⊦ ((i + 1) * (i + 2)) % 2 = 0;%square-even-01: ⊦ (i * (i + 1)) % 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% ⊦ 0 ≤ i → (0 < i * j ↔ 0 < j ∧ i ≠ 0); (s)% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s)% ⊦ ~1 < i ↔ 0 ≤ i; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)% ⊦ 0 < i ∧ 0 < j → 0 < i + j; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j * (j + 1)