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Proof step 42 from proof add-def-any (previous step)

∀ 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, mkDecimal(j1, abs(j) - 1) + mkDecimal(i0, j0) = mkDecimal((j1 * 10) * 10 ^ abs(j0) + i0 * 10 ^ abs(j), abs(j) + abs(j0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 44:
0 < 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, mkDecimal(j1, abs(j) - 1) + mkDecimal(i0, j0) = mkDecimal((j1 * 10) * 10 ^ abs(j0) + i0 * 10 ^ abs(j) -1 * 10, abs(j) + abs(j0))

Rule argument

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

Additional information

Used simplifier rules: abs-06: ⊦ 0 < abs(i) ↔ i ≠ 0; from specification int-abs Used terms: abs(j), 10