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Proof step 19 from proof gcd-mult-pos (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs name: abs-12 seq: ⊦ 0 ≤ i ∧ 0 ≤ j → abs(i + j) = abs(i) + abs(j) subst: [j, i] → [i * i1, j * i1] flags: <all paths>

Additional information

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