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Proof step 3 from proof modmod-ma-negpos-neg (previous step)

j0 + i0 * i < 0, j0 < 0, 0 < j, 0 ≤ i, 0 ≤ i0 ⊦ (j0 + i0 * i % j) % j = (j0 + i0 * i) % j

User applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 5:
i = i1 * j + j1, 0 < j, j0 < 0, j0 + i0 * i < 0, j1 < j, 0 ≤ i0, 0 ≤ i, 0 ≤ j1, 0 ≤ i1 ⊦ (j0 + i0 * j1) % j = (j0 + i0 * i) % j

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, i1, j1] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: i, j, i1, j1