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Proof step 3 from proof modmod-add2-posneg-neg-01 (previous step)

i0 < 0, i + i0 < 0, 0 < j, j + i0 ≤ 0, 0 ≤ i, 0 ≤ i + i0 % j ⊦ (i + i0 % j) % j = 0, (i + i0 % j) % j = (i + i0) % j + j, j = 1

User applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0)
Proof step 5:
i0 = i1 * j + j0, 0 < j, i + i0 < 0, i0 < 0, abs(j0) < j, 0 ≤ i + j0, 0 ≤ i, j + i0 ≤ 0, j0 ≤ 0, i1 ≤ 0 ⊦ (i + j0) % j = (i + i0) % j + j, (i + j0) % j = 0, j = 1

Rule argument

spec: <toplevel> 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] → [i0, j, i1, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s) Used terms: i0, j, i1, j0