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Proof step 44 from proof eq-same (previous step)

i % 10 = 0, mkDecimal(i / 10, abs(j) - 1).int = mkDecimal(i0 / 10, abs(j) - 1).int, mkDecimal(i / 10, abs(j) - 1).scale = mkDecimal(i0 / 10, abs(j) - 1).scale, i0 % 10 = 0, i→n(i) + i→n(i0) ≤ i→n(i / 10) + i→n(i0 / 10), ∀ i1, i2, j. i→n(i1) + i→n(i2) < i→n(i) + i→n(i0) → mkDecimal(i1, j).scale = mkDecimal(i2, j).scale ∧ mkDecimal(i1, j).int = mkDecimal(i2, j).int → i1 = i2 ⊦ j = 0, i = i0, abs(i) / 10 < abs(i) ∧ abs(i0) / 10 < abs(i0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j)
Proof step 46:
10 ≠ 0, i % 10 = 0, mkDecimal(i / 10, abs(j) - 1).int = mkDecimal(i0 / 10, abs(j) - 1).int, mkDecimal(i / 10, abs(j) - 1).scale = mkDecimal(i0 / 10, abs(j) - 1).scale, ∃ j0. i0 = j0 * 10, i→n(i) + i→n(i0) ≤ i→n(i / 10) + i→n(i0 / 10), ∀ i1, i2, j. i→n(i1) + i→n(i2) < i→n(i) + i→n(i0) → mkDecimal(i1, j).scale = mkDecimal(i2, j).scale ∧ mkDecimal(i1, j).int = mkDecimal(i2, j).int → i1 = i2 ⊦ j = 0, i = i0, abs(i) / 10 < abs(i) ∧ abs(i0) / 10 < abs(i0)

Rule argument

spec: int-div inst: name: modex seq: j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j) subst: [j, i] → [10, i0] flags: rewrite

Additional information

Used terms: 10, i0