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Proof step 9 from proof gcd-mult-pos (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ j ≠ 0 → gcd(i, j) = gcd(abs(j), abs(i) % abs(j))
Proof step 11:
1 < j, 0 < i, 0 < i1, i < j, 0 < j, 0 ≤ j, ∀ j0, i0, i2. i→n(j0 * i0 + i2 * i0) < i→n(j * i1 + i * i1) → 0 < i2 ∧ 0 < i0 ∧ 0 ≤ j0 → gcd(i2 * i0, j0 * i0) = gcd(i2, j0) * i0 ⊦ i * i1 = 0, gcd(abs(i * i1), abs(j * i1) % abs(i * i1)) = gcd(i, j) * i1

Rule argument

spec: <toplevel> name: gcd-rec seq: ⊦ j ≠ 0 → gcd(i, j) = gcd(abs(j), abs(i) % abs(j)) subst: [j, i] → [i * i1, j * i1] flags: <all paths>

Additional information

Used simplifier rules: gcd-zero: ⊦ gcd(0, 0) = 0;% ⊦ 0 < i → (i * j = 0 ↔ j = 0); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i * 0 = 0; (s) Used terms: i * i1, j * i1