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Proof step 11 from proof add-def-any (previous step)

∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(i) + i→n(i0) → true → mkDecimal(i1, j) + mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) + i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ mkDecimal (mkDecimal(i, j).int * 10 ^ mkDecimal(i0, j0).scale + mkDecimal(i0, j0).int * 10 ^ mkDecimal(i, j).scale, mkDecimal(i, j).scale + mkDecimal(i0, j0).scale) = mkDecimal(i * 10 ^ abs(j0) + i0 * 10 ^ abs(j), abs(j) + abs(j0)), i0 % 10 = 0 ∧ j0 ≠ 0, i % 10 = 0 ∧ j ≠ 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ ¬(i % 10 = 0 ∧ j ≠ 0 ∧ i ≠ 0) → mkDecimal(i, j).int = i
Proof step 13:
¬(i % 10 = 0 ∧ j ≠ 0 ∧ i ≠ 0), ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(i) + i→n(i0) → true → mkDecimal(i1, j) + mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) + i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ mkDecimal(i * 10 ^ mkDecimal(i0, j0).scale + mkDecimal(i0, j0).int * 10 ^ mkDecimal(i, j).scale, mkDecimal(i, j).scale + mkDecimal(i0, j0).scale) = mkDecimal(i * 10 ^ abs(j0) + i0 * 10 ^ abs(j), abs(j) + abs(j0)), i0 % 10 = 0 ∧ j0 ≠ 0, i % 10 = 0 ∧ j ≠ 0

Rule argument

spec: decimal inst: name: intvalue-base seq: ⊦ ¬(i % 10 = 0 ∧ j ≠ 0 ∧ i ≠ 0) → mkDecimal(i, j).int = i subst: [j, i] → [j, i] flags: rewrite

Additional information

Used terms: j, i