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Proof step 30 from proof gcd-mult-pos (previous step)

0 < i, 0 < i1, 0 ≤ i, j ≤ i, 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 ⊦ j * i1 = 0, gcd(abs(j * i1), abs(i * i1) % abs(j * i1)) = gcd(i, j) * i1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 31:
0 < i1, 0 < i, 0 < j, j ≤ i, 0 ≤ i, ∀ 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 % j * i1) = gcd(i, j) * i1, i1 = 0, j = 0

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ i * j = 0 ↔ i = 0 ∨ j = 0; (ws)%0 ≠ j, i0 ≠ 0 ⊦ (i0 * i) % (i0 * j) = i % j * i0; (s)% ⊦ i * 0 = 0; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ 0 < j → abs(j) = j; (s)% ⊦ 0 ≤ i → abs(i) = i; (s)% ⊦ abs(i * j) = abs(i) * abs(j); (s)