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Proof step 40 from proof sub-def-any (previous step)

mkDecimal(i, j) - mkDecimal(j1, abs(j0) - 1) = mkDecimal((i * 10 ^ (abs(j0) - 1) - j1 * 10 ^ abs(j)) * 10, abs(j) + abs(j0)), 0 < abs(j) + abs(j0), ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(i) + i→n(j1 * 10) → true → mkDecimal(i1, j) - mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) - i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ j0 = 0, i * 10 ^ abs(j0) = (i * 10 ^ (abs(j0) - 1) - j1 * 10 ^ abs(j)) * 10 + (j1 * 10) * 10 ^ abs(j), i % 10 = 0 ∧ j ≠ 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 41:
⊦ (i - j) * j0 = i * j0 - j * j0
Proof step 42:
mkDecimal(i, j) - mkDecimal(j1, abs(j0) - 1) = mkDecimal((i * 10 ^ (abs(j0) - 1)) * 10 - (j1 * 10 ^ abs(j)) * 10, abs(j) + abs(j0)), 0 < abs(j) + abs(j0), ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(i) + i→n(j1 * 10) → true → mkDecimal(i1, j) - mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) - i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ j0 = 0, i * 10 ^ abs(j0) = ((i * 10 ^ (abs(j0) - 1)) * 10 - (j1 * 10 ^ abs(j)) * 10) + (j1 * 10) * 10 ^ abs(j), i % 10 = 0 ∧ j ≠ 0

Rule argument

spec: int-mult inst: name: dist-sub-split seq: ⊦ (i - j) * j0 = i * j0 - j * j0 subst: [j0, j, i] → [10, j1 * 10 ^ abs(j), i * 10 ^ (abs(j0) - 1)] flags: rewrite

Additional information

Used terms: 10, j1 * 10 ^ abs(j), i * 10 ^ (abs(j0) - 1)