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Proof step 16 from proof divide-by-10 (previous step)

0 ≤ j, 0 ≤ i0, j < 10, i0 * 10 + j ≤ 65536 ⊦ (8 * i0 * 2 ^ 16 + i0 * (2 * 2 ^ 16 - 10 - 10 * 13106) + j * (2 ^ 16 - 1 - 13106)) / 524288 = i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 18:
¬((8 * i0 * 2 ^ 16) % 524288 = 0 ∧ 0 ≤ 8 * i0 * 2 ^ 16 ∧ 0 ≤ i0 * (2 * 2 ^ 16 - 10 - 10 * 13106) + j * (2 ^ 16 - 1 - 13106) ∧ 0 < 524288), 0 ≤ j, 0 ≤ i0, j < 10, i0 * 10 + j ≤ 65536 ⊦ (8 * i0 * 2 ^ 16 + i0 * (2 * 2 ^ 16 - 10 - 10 * 13106) + j * (2 ^ 16 - 1 - 13106)) / 524288 = i0
Proof step 21:
(8 * i0 * 2 ^ 16) % 524288 = 0 ∧ 0 ≤ 8 * i0 * 2 ^ 16 ∧ 0 ≤ i0 * (2 * 2 ^ 16 - 10 - 10 * 13106) + j * (2 ^ 16 - 1 - 13106) ∧ 0 < 524288, 0 ≤ j, 0 ≤ i0, j < 10, i0 * 10 + j ≤ 65536 ⊦ (8 * i0 * 2 ^ 16) / 524288 + (i0 * (2 * 2 ^ 16 - 10 - 10 * 13106) + j * (2 ^ 16 - 1 - 13106)) / 524288 = i0

Rule argument

spec: int-div inst: name: div-add1-pospos seq: ⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [524288, i0 * (2 * 2 ^ 16 - 10 - 10 * 13106) + j * (2 ^ 16 - 1 - 13106), 8 * i0 * 2 ^ 16] flags: rewrite

Additional information

Used terms: 524288, i0 * (2 * 2 ^ 16 - 10 - 10 * 13106) + j * (2 ^ 16 - 1 - 13106), 8 * i0 * 2 ^ 16