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Proof step 21 from proof gcd-mult-pos (previous step)

i < j, 0 < i1, 0 < i, 0 ≤ j, 1 < j, 0 < j, 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 < 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 insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: abs-12 seq: ⊦ 0 ≤ i ∧ 0 ≤ j → abs(i + j) = abs(i) + abs(j) subst: [j, i] → [i * i1, j % i * i1] flags: rewrite

Additional information

Used simplifier rules: zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%mod-zero: ⊦ 0 < j ∧ 0 ≤ i → 0 ≤ i % j; from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs Used terms: i * i1, j % i * i1