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Proof step 27 from proof gcdhex (previous step)

gcd(i, j) = j * i0 + (i * j0 - (i / j * j) * j0), gcd(i, j) = gcd(j, i - i / j * j), 0 < j, 0 ≤ i, j < i, 1 < i, 0 < i, ∀ i0, j. i→n(i0) < i→n(i) → 0 ≤ i0 ∧ 0 ≤ j ∧ j ≤ i0 → (∃ i1, j0. gcd(i0, j) = i0 * i1 + j * j0) ⊦ j * i0 + (i * j0 - (i / j * j) * j0) = i * j0 + j * (i0 - i / j * j0), i = j, j = 0, ∃ i0, j0. gcd(i, j) = i * i0 + j * j0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: dist-sub-split seq: ⊦ (i - j) * j0 = i * j0 - j * j0 subst: [j0, j, i] → [j, i / j * j0, i0] flags: rewrite

Additional information

Used simplifier rules: sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%Sub-def: ⊦ i + (j - i) = j; from specification int-abs%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j, i / j * j0, i0