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Proof step 32 from proof le-abs-numerator (previous step)

i / j < 0, j < 0, i + i / j < 0, 0 ≤ i ⊦ j = 0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 33:
⊦ j < 0 ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j) ∧ i0 ≤ 0)
Proof step 34:
i = i0 * j + j0, i + i0 < 0, j < 0, i0 < 0, j0 < abs(j), 0 ≤ i, 0 ≤ j0, i0 ≤ 0 ⊦ j = 0

Rule argument

spec: <toplevel> name: div-elim-posneg seq: ⊦ j < 0 ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j) ∧ i0 ≤ 0) subst: [i, j, i0, j0] → [i, j, i0, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: i, j, i0, j0