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Proof step 11 from proof div-eq-same (previous step)

j * i / j = i ⊦ 0 = j, i % j = 0, j < 0 ∧ 0 ≤ i, 0 < j ∧ i < 0, j < 0 ∧ i < 0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 12:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 13:
j * i / j = i ⊦ 0 < j ∧ 0 ≤ i, 0 = j, i % j = 0, j < 0 ∧ 0 ≤ i, 0 < j ∧ i < 0, j < 0 ∧ i < 0

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] → [i, j, i0, j0] flags: rewrite, current

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult%eq: ⊦ i = i + j ↔ j = 0; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i, j, i0, j0