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Proof step 9 from proof mkDecimal-maxScale-le-int (previous step)

i % 10 = 0, maxScale(i) ≤ abs(j), ∀ i0, j. i→n(i0) < i→n(i) → maxScale(i0) ≤ abs(j) → mkDecimal(i0, j).int = i0 / 10 ^ maxScale(i0) ⊦ j = 0, i = 0, mkDecimal(i / 10, abs(j) - 1).int = i / 10 ^ maxScale(i)

User applied rule apply induction

And obtained the following new goals

Proof step 10:
i % 10 = 0, maxScale(i) ≤ abs(j), ∀ i0, j. i→n(i0) < i→n(i) → maxScale(i0) ≤ abs(j) → mkDecimal(i0, j).int = i0 / 10 ^ maxScale(i0) ⊦ maxScale(i / 10) ≤ abs(abs(j) - 1), j = 0, i = 0, mkDecimal(i / 10, abs(j) - 1).int = i / 10 ^ maxScale(i)
Proof step 16:
mkDecimal(i / 10, abs(j) - 1).int = (i / 10) / 10 ^ maxScale(i / 10), i % 10 = 0, maxScale(i) ≤ abs(j), ∀ i0, j. i→n(i0) < i→n(i) → maxScale(i0) ≤ abs(j) → mkDecimal(i0, j).int = i0 / 10 ^ maxScale(i0) ⊦ j = 0, i = 0, mkDecimal(i / 10, abs(j) - 1).int = i / 10 ^ maxScale(i)

Rule argument

subst: [i0, j] → [i / 10, abs(j) - 1]

Additional information

Used simplifier rules: labs: ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j); from specification intnat%abs: j ≠ 0 ⊦ abs(i / j) = abs(i) / abs(j); from specification int-div%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%abs-div10-le: ⊦ abs(i) ≤ abs(i) / 10 ↔ i = 0; from specification int-div Used terms: i / 10, abs(j) - 1