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Proof step 9 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(i, j) + mkDecimal(i0, j0) = 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 10:
⊦ dec_i + dec_j = mkDecimal(dec_i.int * 10 ^ dec_j.scale + dec_j.int * 10 ^ dec_i.scale, dec_i.scale + dec_j.scale)
Proof step 11:
∀ 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

Rule argument

spec: inst: name: add-def seq: ⊦ dec_i + dec_j = mkDecimal (dec_i.int * 10 ^ dec_j.scale + dec_j.int * 10 ^ dec_i.scale, dec_i.scale + dec_j.scale) subst: [dec_j, dec_i] → [mkDecimal(i0, j0), mkDecimal(i, j)] flags: rewrite, current

Additional information

Used terms: mkDecimal(i0, j0), mkDecimal(i, j)