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Proof step 14 from proof modmod-mult (previous step)

(i1 * (j1 + i2 * j) + j0 * i2) * j + j0 * j1 < 0, 0 < j, j0 < j, j1 < j, 0 ≤ i2, 0 ≤ j0, 0 ≤ j1, 0 ≤ i1 ⊦ (j0 * j1) % j = ((i1 * (j1 + i2 * j) + j0 * i2) * j + j0 * j1) % j

User applied rule insert elim lemma

And obtained the following new goals

Proof step 15:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j)
Proof step 16:
j1 < j, j0 < j, 0 < j, (i1 * (j1 + i2 * j) + j0 * i2) * j + j0 * j1 < 0, 0 ≤ i1, 0 ≤ j1, 0 ≤ j0, 0 ≤ i2 ⊦ (j0 * j1) % j = ((i1 * (j1 + i2 * j) + j0 * i2) * j + j0 * j1) % j, 0 < j ∧ j0 * j1 < 0

Rule argument

spec: <toplevel> name: div-elim2 seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j) subst: [i, j, i0, j0] → [j0 * j1, j, i, j2]

Additional information

Used simplifier rules: ⊦ 0 ≤ i → (i * j < 0 ↔ j < 0 ∧ i ≠ 0); (s)% ⊦ i ≤ j → ¬ j < i; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j0 * j1, j, i, j2