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Proof step 3 from proof diveq-neg (previous step)

abs(j0) < j, i0 * j + j0 < 0, i0 < 0, j0 ≤ 0 ⊦ (i0 * j + j0) / j = i0

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)
Proof step 5:
i0 * j + j0 = i1 * j + j1, i0 < 0, i0 * j + j0 < 0, abs(j0) < j, abs(j1) < j, 0 < j, j0 ≤ 0, j1 ≤ 0 ⊦ i1 = i0

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] → [i0 * j + j0, j, i, j1] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs% ⊦ ¬ abs(i) < 0; (s) Used terms: i0 * j + j0, j, i, j1