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Proof step 2 from proof square-inject (previous step)

((i + j) * (i + j + 1)) / 2 + j = ((i0 + j0) * (i0 + j0 + 1)) / 2 + j0, 0 ≤ i, 0 ≤ j, 0 ≤ i0, 0 ≤ j0 ⊦ i = i0 ∧ j = j0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 3:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 4:
(i0 + j0) * (i0 + j0 + 1) = i1 * 2 + j1 ∧ 0 ≤ j1 ∧ j1 < 2 ∧ 0 ≤ i1 ∧ 0 < 2 ∧ 0 ≤ (i0 + j0) * (i0 + j0 + 1), ((i + j) * (i + j + 1)) / 2 + j = i1 + j0, 0 ≤ i, 0 ≤ j, 0 ≤ i0, 0 ≤ j0 ⊦ i = i0 ∧ j = j0

Rule argument

spec: int-div inst: name: div-elim-pospos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [(i0 + j0) * (i0 + j0 + 1), 2, i1, j1] flags: rewrite, current

Additional information

Used simplifier rules: lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%square-zero-02: ⊦ 0 ≤ j ∧ 0 ≤ i → ¬ (i + j) * (i + j + 1) < 0; from specification int-mult%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%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: (i0 + j0) * (i0 + j0 + 1), 2, i1, j1