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Proof step 12 from proof mod-neq (previous step)

i % j = j ⊦ j = 0, j < 0 ∧ 0 ≤ i, 0 < j ∧ i < 0, j < 0 ∧ i < 0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 13:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 14:
i % j = j ⊦ j = 0, j < 0 ∧ i < 0, 0 < j ∧ i < 0, j < 0 ∧ 0 ≤ i, 0 < j ∧ 0 ≤ i

Rule argument

spec: <toplevel> name: div-elim-pospos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [i, j, i0, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ ¬ i < i; (s) Used terms: i, j, i0, j0