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Proof step 31 from proof add-def-any (previous step)

∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(i) + i→n(j1 * 10) → true → mkDecimal(i1, j) + mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) + i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ j0 = 0, mkDecimal(i, j) + mkDecimal(j1, abs(j0) - 1) = mkDecimal(10 * (i * 10 ^ (abs(j0) - 1) + j1 * 10 ^ abs(j)), abs(j) + abs(j0)), i % 10 = 0 ∧ j ≠ 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 32:
⊦ i % 10 = 0 ∧ j ≠ 0 → mkDecimal(i, j) = mkDecimal(i / 10, abs(j) - 1)
Proof step 33:
(10 * (i * 10 ^ (abs(j0) - 1) + j1 * 10 ^ abs(j))) % 10 = 0 ∧ abs(j) + abs(j0) ≠ 0, ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(i) + i→n(j1 * 10) → true → mkDecimal(i1, j) + mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) + i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ j0 = 0, mkDecimal(i, j) + mkDecimal(j1, abs(j0) - 1) = mkDecimal((10 * (i * 10 ^ (abs(j0) - 1) + j1 * 10 ^ abs(j))) / 10, abs(abs(j) + abs(j0)) - 1), i % 10 = 0 ∧ j ≠ 0

Rule argument

spec: decimal inst: name: mkDecimal-rec seq: ⊦ i % 10 = 0 ∧ j ≠ 0 → mkDecimal(i, j) = mkDecimal(i / 10, abs(j) - 1) subst: [j, i] → [abs(j) + abs(j0), 10 * (i * 10 ^ (abs(j0) - 1) + j1 * 10 ^ abs(j))] flags: rewrite

Additional information

Used simplifier rules: mod-zero-num: i0 % j = 0, j ≠ 0 ⊦ (i * (* i0)) % (* j) = 0; from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%abs-add-zero: ⊦ abs(i) + abs(j) = 0 ↔ i = 0 ∧ j = 0; 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 Used terms: abs(j) + abs(j0), 10 * (i * 10 ^ (abs(j0) - 1) + j1 * 10 ^ abs(j))