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Proof step 72 from proof sub-def-any (previous step)

mkDecimal(j1, abs(j) - 1) - mkDecimal(i0, j0) = mkDecimal((j1 * 10 ^ abs(j0) - i0 * 10 ^ (abs(j) - 1)) * 10, abs(j0) + abs(j)), 0 < abs(j0) + abs(j), ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(j1 * 10) + i→n(i0) → true → mkDecimal(i1, j) - mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) - i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ j = 0, (j1 * 10) * 10 ^ abs(j0) = (j1 * 10 ^ abs(j0) - i0 * 10 ^ (abs(j) - 1)) * 10 + i0 * 10 ^ abs(j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 73:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 74:
¬ 0 < abs(j0), mkDecimal(j1, abs(j) - 1) - mkDecimal(i0, j0) = mkDecimal((j1 * 10 ^ abs(j0) - i0 * 10 ^ (abs(j) - 1)) * 10, abs(j0) + abs(j)), 0 < abs(j0) + abs(j), ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(j1 * 10) + i→n(i0) → true → mkDecimal(i1, j) - mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) - i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ j = 0, (j1 * 10) * 10 ^ abs(j0) = (j1 * 10 ^ abs(j0) - i0 * 10 ^ (abs(j) - 1)) * 10 + i0 * 10 ^ abs(j)
Proof step 80:
0 < abs(j0), mkDecimal(j1, abs(j) - 1) - mkDecimal(i0, j0) = mkDecimal((j1 * 10 ^ abs(j0) -1 * 10 - i0 * 10 ^ (abs(j) - 1)) * 10, abs(j0) + abs(j)), 0 < abs(j0) + abs(j), ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(j1 * 10) + i→n(i0) → true → mkDecimal(i1, j) - mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) - i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ j = 0, (j1 * 10) * 10 ^ abs(j0) -1 * 10 = (j1 * 10 ^ abs(j0) -1 * 10 - i0 * 10 ^ (abs(j) - 1)) * 10 + i0 * 10 ^ abs(j)

Rule argument

spec: int-pot inst: name: Pot-pos seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [abs(j0), 10] flags: rewrite

Additional information

Used terms: abs(j0), 10