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Proof step 41 from proof gcd-mult-pos (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: Abs-pos seq: ⊦ ¬ j < 0 → abs(j) = j subst: [j] → [i % j] flags: rewrite

Additional information

Used simplifier rules: lst-01: ⊦ ¬ i < j → (i ≠ j ↔ j < i); from specification int-basic1%mod-les: ⊦ 0 < j → ¬ j ≤ i % j; from specification int-div%mod-zero-ls: ⊦ 0 < i ∧ 0 < j → ¬ i % j < 0; from specification int-div Used terms: i % j