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Proof step 43 from proof equality-any (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 44:
⊦ 0 < j ∧ 0 ≤ j0 ∧ j0 ≤ j1 → (i * j ^ j0 = i0 * j ^ j1 ↔ i = i0 * j ^ (j1 - j0))
Proof step 45:
¬(0 < 10 ∧ 0 ≤ abs(j0) ∧ abs(j0) ≤ abs(j)), i * 10 ^ abs(j0) = i0 * 10 ^ abs(j), ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(i) + i→n(i0) → true → (mkDecimal(i1, j) = mkDecimal(i2, j0) ↔ i1 * 10 ^ abs(j0) = i2 * 10 ^ abs(j)) ⊦ i = i0 ∧ abs(j) = abs(j0), i % 10 = 0, i0 % 10 = 0, i = 0, i0 = 0
Proof step 55:
0 < 10 ∧ 0 ≤ abs(j0) ∧ abs(j0) ≤ abs(j), i = i0 * 10 ^ (abs(j) - abs(j0)), ∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(i) + i→n(i0) → true → (mkDecimal(i1, j) = mkDecimal(i2, j0) ↔ i1 * 10 ^ abs(j0) = i2 * 10 ^ abs(j)) ⊦ i = i0 ∧ abs(j) = abs(j0), i % 10 = 0, i0 % 10 = 0, i = 0, i0 = 0

Rule argument

spec: int-pot inst: name: pot-eq-sub seq: ⊦ 0 < j ∧ 0 ≤ j0 ∧ j0 ≤ j1 → (i * j ^ j0 = i0 * j ^ j1 ↔ i = i0 * j ^ (j1 - j0)) subst: [j1, i0, j0, j, i] → [abs(j), i0, abs(j0), 10, i] flags: rewrite

Additional information

Used terms: abs(j), i0, abs(j0), 10, i