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Proof step 125 from proof shiftright-rec-8-neg (previous step)

(i0 - 1) >> (j - 1) = ((i0 - 1) / 256 - 1) >> (j - 9), i0 < j, i0 ≤ 0, 0 < j, 8 < j, ∀ i, j0. i→n(j0) < i→n(j) → 8 ≤ j0 ∧ i < 0 ∧ 0 < j0 ∧ 0 ≤ j0 → i >> j0 = (i - 255) / 256 >> (j0 - 8) ⊦ j = 0, (i0 - 1) / 256 = (i0 * 2 - 1) / (2 * 256), (i0 * 2 - 1) % 256 = 0, j = 8, (i0 - 1) % 256 = 0, (i0 - 1) >> (j - 1) = ((i0 * 2 - 1) / (2 * 256) - 1) >> (j - 9)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 126:
⊦ j ≠ 0 ∧ j0 ≠ 0 → i / (j * j0) = (i / j) / j0
Proof step 127:
2 ≠ 0 ∧ 256 ≠ 0, (i0 - 1) >> (j - 1) = ((i0 - 1) / 256 - 1) >> (j - 9), i0 < j, i0 ≤ 0, 0 < j, 8 < j, ∀ i, j0. i→n(j0) < i→n(j) → 8 ≤ j0 ∧ i < 0 ∧ 0 < j0 ∧ 0 ≤ j0 → i >> j0 = (i - 255) / 256 >> (j0 - 8) ⊦ j = 0, (i0 - 1) / 256 = ((i0 * 2 - 1) / 2) / 256, (i0 * 2 - 1) % 256 = 0, j = 8, (i0 - 1) % 256 = 0, (i0 - 1) >> (j - 1) = (((i0 * 2 - 1) / 2) / 256 - 1) >> (j - 9)

Rule argument

seq: ⊦ j ≠ 0 ∧ j0 ≠ 0 → i / (j * j0) = (i / j) / j0 subst: [j0, j, i] → [256, 2, i0 * 2 - 1] rotate: F spec: int-div inst: name: div-div-intro

Additional information

Used terms: 256, 2, i0 * 2 - 1