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Proof step 12 from proof mod-pot-shift-pos (previous step)

0 < j, 0 ≤ i2, ∀ i1, j, i. i→n(i1) < i→n(i2 + 1) → 0 < j ∧ 0 < i1 → (i % j) ^ i1 % j = i ^ i1 % j ∨ i1 = 0 ⊦ ((i % j) ^ i2 * i % j) % j = (i ^ i2 * i) % j, ~1 = i2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
0 ≠ j ⊦ (i * i0) % j = (i % j * i0 % j) % j
Proof step 14:
0 < j, 0 ≤ i2, ∀ i1, j, i. i→n(i1) < i→n(i2 + 1) → 0 < j ∧ 0 < i1 → (i % j) ^ i1 % j = i ^ i1 % j ∨ i1 = 0 ⊦ ((i % j) ^ i2 % j * (i % j) % j) % j = (i ^ i2 * i) % j, ~1 = i2, 0 = j

Rule argument

spec: int-div name: modmod-law-mult seq: 0 ≠ j ⊦ (i * i0) % j = (i % j * i0 % j) % j subst: [j, i0, i] → [j, i % j, (i % j) ^ i2] flags: <all paths>

Additional information

Used terms: j, i % j, (i % j) ^ i2