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Proof step 56 from proof shiftright-rec-8-neg (previous step)

j0 * 128 >> (j - 1) = (j0 * 128) / 256 >> (j - 9), (j0 * 128) % 256 = 0, 8 < j, 0 < j, j0 < 0, ∀ i, j0. i→n(j0) < i→n(j) → 8 ≤ j0 ∧ i < 0 ∧ 0 < j0 ∧ 0 ≤ j0 → i >> j0 = (i - 255) / 256 >> (j0 - 8) ⊦ j = 8, j0 * 128 >> (j - 1) = (j0 - 1) / 2 >> (j - 9), j = 0, (j0 * 128) / 256 = (j0 - 1) / 2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 57:
j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j)
Proof step 58:
256 ≠ 0, j0 * 128 >> (j - 1) = (j0 * 128) / 256 >> (j - 9), ∃ j1. j0 * 128 = j1 * 256, 8 < j, 0 < j, j0 < 0, ∀ i, j0. i→n(j0) < i→n(j) → 8 ≤ j0 ∧ i < 0 ∧ 0 < j0 ∧ 0 ≤ j0 → i >> j0 = (i - 255) / 256 >> (j0 - 8) ⊦ j = 8, j0 * 128 >> (j - 1) = (j0 - 1) / 2 >> (j - 9), j = 0, (j0 * 128) / 256 = (j0 - 1) / 2

Rule argument

seq: j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j) subst: [j, i] → [256, j0 * 128] rotate: F spec: int-div inst: name: modex

Additional information

Used terms: 256, j0 * 128