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Proof step 6 from proof modex (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 7:
⊦ j < 0 ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j))
Proof step 8:
i % j = 0 ⊦ j = 0, ∃ j0. i = j0 * j, 0 < j ∧ 0 ≤ i, j < 0 ∧ 0 ≤ i

Rule argument

spec: <toplevel> name: div-elim1 seq: ⊦ j < 0 ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < abs(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% ⊦ i < 0 → (0 ≤ i * j ↔ j ≤ 0); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ 0 < ~ i ↔ i < 0; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i < j → ¬ j < i; (s)% ⊦ i < 0 → (i * j = i * j0 ↔ j = j0); (s) Used terms: i, j, i0, j1