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Proof step 27 from proof div-div-pred-01 (previous step)

j0 ≤ 0, i1 * 4 + j1 ≤ ~3, i1 * 4 + j1 < 0, abs(j0) < 2, abs(j1) < 4, i1 ≤ 0, j1 ≤ 0, 8 * i1 + j0 + j1 * 2 < ~7, 8 * i1 + j0 + j1 * 2 < 0 ⊦ i1 = (8 * i1 + j0 + j1 * 2) / 8

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ i % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 29:
¬((8 * i1) % 8 = 0 ∧ 8 * i1 < 0 ∧ j0 + j1 * 2 < 0 ∧ 0 < 8), j0 ≤ 0, i1 * 4 + j1 ≤ ~3, i1 * 4 + j1 < 0, abs(j0) < 2, abs(j1) < 4, i1 ≤ 0, j1 ≤ 0, 8 * i1 + j0 + j1 * 2 < ~7, 8 * i1 + j0 + j1 * 2 < 0 ⊦ i1 = (8 * i1 + j0 + j1 * 2) / 8
Proof step 36:
(8 * i1) % 8 = 0 ∧ 8 * i1 < 0 ∧ j0 + j1 * 2 < 0 ∧ 0 < 8, j0 ≤ 0, i1 * 4 + j1 ≤ ~3, i1 * 4 + j1 < 0, abs(j0) < 2, abs(j1) < 4, i1 ≤ 0, j1 ≤ 0, 8 * i1 + j0 + j1 * 2 < ~7, 8 * i1 + j0 + j1 * 2 < 0 ⊦ i1 = (8 * i1) / 8 + (j0 + j1 * 2) / 8

Rule argument

spec: inst: name: div-add1-negneg seq: ⊦ i % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [8, j0 + j1 * 2, 8 * i1] flags: rewrite, current

Additional information

Used terms: 8, j0 + j1 * 2, 8 * i1