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Proof step 11 from proof square-interval (previous step)

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

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-02: ⊦ ((i + 1) * (i + 2)) % 2 = 0;%square-even: ⊦ 0 ≤ i → (i * (i + 1)) % 2 = 0;%mult-ls-01: ⊦ 0 < j ∧ i0 < i / j → i0 * j < i;%div-lezero: ⊦ 0 < j → (i / j ≤ 0 ↔ i < j);%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 ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ 0 + i = i; (s)% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ 0 < i → ¬ i * (i + 1) < 2; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j * (j + 1)