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Proof step 15 from proof modex (previous step)

i % j = 0, i < 0, j < 0 ⊦ ∃ j0. i = j0 * j

User applied rule insert elim lemma

And obtained the following new goals

Proof step 16:
⊦ j < 0 ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ j < j0)

Rule argument

spec: <toplevel> name: div-elim3 seq: ⊦ j < 0 ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ j < j0) subst: [i, j, i0, j0] → [i, j, i0, j1]

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i + 0 = i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i < 0 → (i * j < 0 ↔ 0 < j); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i < 0 → (i * j = i * j0 ↔ j = j0); (s) Used terms: i, j, i0, j1