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Proof step 15 from proof mkDecimal-enlarge (previous step)

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

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [j1, i, j] → [j0 - 1, i, j]

Additional information

Used simplifier rules: sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-zero: ⊦ i + 0 = i; from specification int-basic2%abs-01: ⊦ abs(abs(i)) = abs(i); from specification int-abs%abs-07: ⊦ abs(i) = 0 ↔ i = 0; from specification int-abs%abs-pred: ⊦ i→n(i - 1) < i→n(i) ↔ 0 < i; from specification intnat Used terms: j0 - 1, i, j