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Proof step 8 from proof div-add1-negpos-neg-01 (previous step)

i1 * j < j0, i1 * j + i2 * j < j, i1 * j + i2 * j < j0, (i1 * j + i2 * j) + 1 < j0, 2 + i1 < j, i1 < j0, 0 ≤ i2, 0 < j, 1 < j, 1 + i1 * j + i2 * j + j0 < j, j + i1 * j ≤ 0, i1 < 0, i1 < j, j0 < j, 0 < j0, i1 * j + i2 * j < ~1, i1 * j + i2 * j + j0 < j, i1 * j + i2 * j + j0 < 0, 0 ≤ i2 * j + j0, j + i1 * j < j0, i1 + 1 < j0, i1 * j + i2 * j < 0, 2 + i1 * j + i2 * j < j ⊦ j0 = 0, j = 1, (i1 * j + i2 * j + j0) / j = i1 + i2 + 1

User applied rule insert elim lemma

And obtained the following new goals

Proof step 9:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0)
Proof step 10:
i1 * j + i2 * j + j0 = i0 * j + j1 ∧ j1 ≤ 0 ∧ abs(j1) < j ∧ i0 ≤ 0, i1 * j < j0, i1 * j + i2 * j < j, i1 * j + i2 * j < j0, (i1 * j + i2 * j) + 1 < j0, 2 + i1 < j, i1 < j0, 0 ≤ i2, 0 < j, 1 < j, 1 + i1 * j + i2 * j + j0 < j, j + i1 * j ≤ 0, i1 < 0, i1 < j, j0 < j, 0 < j0, i1 * j + i2 * j < ~1, i1 * j + i2 * j + j0 < j, i1 * j + i2 * j + j0 < 0, 0 ≤ i2 * j + j0, j + i1 * j < j0, i1 + 1 < j0, i1 * j + i2 * j < 0, 2 + i1 * j + i2 * j < j ⊦ j0 = 0, j = 1, i0 = i1 + i2 + 1

Rule argument

spec: int-div inst: name: div-elim-negpos seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0) subst: [i, j, i0, j0] → [i1 * j + i2 * j + j0, j, i, j1] flags: current, []

Additional information

Used simplifier rules: add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs Used terms: i1 * j + i2 * j + j0, j, i, j1