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Proof step 53 from proof equality-any (previous step)

abs(j) < abs(j0), ∀ i1, i2, j2, j1. i→n(i1) + i→n(i2) < i→n(i) + i→n(i * 10 ^ (abs(j0) - abs(j))) → true → (mkDecimal(i1, j2) = mkDecimal(i2, j1) ↔ i1 * 10 ^ abs(j1) = i2 * 10 ^ abs(j2)) ⊦ i % 10 = 0, (i * 10 ^ (abs(j0) - abs(j))) % 10 = 0, i = 0, abs(j) = abs(j0), 10 ^ (abs(j0) - abs(j)) % 10 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-pot inst: name: Pot-pos seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [abs(j0) - abs(j), 10] flags: rewrite

Additional information

Used simplifier rules: sub-pred: ⊦ i -1 = i - 1; from specification int-abs%sub-sub: ⊦ i - j - i0 = i - j + i0; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%mod-zero-num: i0 % j = 0, j ≠ 0 ⊦ (i * (* i0)) % (* j) = 0; from specification int-div%commutativity: ⊦ i * j = j * i; from specification int-mult%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%add-zero: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: abs(j0) - abs(j), 10