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Proof step 16 from proof gcd-mult-pos (previous step)

i < j, 0 < i1, 0 < i, 0 ≤ j, 1 < j, 0 < j, ∀ j0, i0, i2. i→n(j0 * i0 + i2 * i0) < i→n(j * i1 + i * i1) → 0 < i0 ∧ 0 < i2 ∧ 0 ≤ j0 → gcd(i2 * i0, j0 * i0) = gcd(i2, j0) * i0 ⊦ i1 = 0, i = 0, gcd(i * i1, j % i * i1) = gcd(i, j % i) * i1

User applied rule apply induction

And obtained the following new goals

Proof step 17:
i < j, 0 < i1, 0 < i, 0 ≤ j, 1 < j, 0 < j, ∀ j0, i0, i2. i→n(j0 * i0 + i2 * i0) < i→n(j * i1 + i * i1) → 0 < i0 ∧ 0 < i2 ∧ 0 ≤ j0 → gcd(i2 * i0, j0 * i0) = gcd(i2, j0) * i0 ⊦ i→n(j % i * i1 + i * i1) < i→n(j * i1 + i * i1), i1 = 0, i = 0, gcd(i * i1, j % i * i1) = gcd(i, j % i) * i1

Rule argument

subst: [j0, i0, i2] → [j % i, i1, i]

Additional information

Used simplifier rules: mod-zero: ⊦ 0 < j ∧ 0 ≤ i → 0 ≤ i % j; from specification int-div Used terms: j % i, i1, i