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Proof step 19 from proof gcdhex (previous step)

gcd(i, j) = j * i0 + i % j * j0, gcd(i, j) = gcd(j, i % j), 0 < j, 0 ≤ i, j < i, 1 < i, 0 < i, ∀ i0, j. i→n(i0) < i→n(i) → 0 ≤ i0 ∧ 0 ≤ j ∧ j ≤ i0 → (∃ i1, j0. gcd(i0, j) = i0 * i1 + j * j0) ⊦ i = j, j = 0, ∃ i0, j0. gcd(i, j) = i * i0 + j * j0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
j ≠ 0 ⊦ i % j = i - i / j * j
Proof step 21:
j ≠ 0, gcd(i, j) = j * i0 + (i - i / j * j) * j0, gcd(i, j) = gcd(j, i - i / j * j), 0 < j, 0 ≤ i, j < i, 1 < i, 0 < i, ∀ i0, j. i→n(i0) < i→n(i) → 0 ≤ i0 ∧ 0 ≤ j ∧ j ≤ i0 → (∃ i1, j0. gcd(i0, j) = i0 * i1 + j * j0) ⊦ i = j, j = 0, ∃ i0, j0. gcd(i, j) = i * i0 + j * j0

Rule argument

spec: int-div inst: name: mod-rew seq: j ≠ 0 ⊦ i % j = i - i / j * j subst: [j, i] → [j, i] flags: rewrite

Additional information

Used terms: j, i