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Proof step 9 from proof modex (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j)
Proof step 11:
i % j = 0 ⊦ j = 0, ∃ j0. i = j0 * j, 0 < j ∧ 0 ≤ i, j < 0 ∧ 0 ≤ i, 0 < j ∧ i < 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] → [i, j, i0, j1] flags: <rotated> <all paths>

Additional information

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