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Proof step 37 from proof gcd-mult-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%abs-eqall: ⊦ abs(i * j) = abs(i) * abs(j); from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult