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Proof step 31 from proof square-sqrt (previous step)

mkDecimal(sqrt(mkDecimal(j * j, 2 * (abs(i0) - 1)).int), mkDecimal(j * j, 2 * (abs(i0) - 1)).scale / 2) = mkDecimal(j, abs(i0) - 1), 0 < j, ∀ i1, i0. i→n(i1) < i→n(j * 10) → 0 ≤ i1 → mkDecimal(sqrt(mkDecimal(i1 * i1, 2 * abs(i0)).int), mkDecimal(i1 * i1, 2 * abs(i0)).scale / 2) = mkDecimal(i1, i0) ⊦ j = 0, i0 = 0, mkDecimal(sqrt(mkDecimal(j * j, 2 * abs(i0) - 2).int), mkDecimal(j * j, 2 * abs(i0) - 2).scale / 2) = mkDecimal(j * 10, i0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 32:
⊦ (i - j) * j0 = i * j0 - j * j0
Proof step 33:
mkDecimal(sqrt(mkDecimal(j * j, abs(i0) * 2 - 1 * 2).int), mkDecimal(j * j, abs(i0) * 2 - 1 * 2).scale / 2) = mkDecimal(j, abs(i0) - 1), 0 < j, ∀ i1, i0. i→n(i1) < i→n(j * 10) → 0 ≤ i1 → mkDecimal(sqrt(mkDecimal(i1 * i1, 2 * abs(i0)).int), mkDecimal(i1 * i1, 2 * abs(i0)).scale / 2) = mkDecimal(i1, i0) ⊦ j = 0, i0 = 0, mkDecimal(sqrt(mkDecimal(j * j, 2 * abs(i0) - 2).int), mkDecimal(j * j, 2 * abs(i0) - 2).scale / 2) = mkDecimal(j * 10, i0)

Rule argument

spec: int-mult inst: name: dist-sub-split seq: ⊦ (i - j) * j0 = i * j0 - j * j0 subst: [j0, j, i] → [2, 1, abs(i0)] flags: rewrite

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: 2, 1, abs(i0)