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Proof step 19 from proof modmod-add-zero (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 20:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 21:
(i + j0) % j = 0, i0 = i1 * j + j0, 0 < j, j0 < j, 0 ≤ i0, 0 ≤ j0, 0 ≤ i1 ⊦ (i + i0) % j = 0, 0 = j, 0 ≤ i, i ≤ 0 ∧ i0 ≤ 0

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] → [i0, j, i1, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i ≤ j ↔ ¬ j < i; (ss) Used terms: i0, j, i1, j0