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Proof step 15 from proof cast-law-add-pos (previous step)

0 ≤ j, 0 ≤ j0, 0 < i, 2 ^ (i - 1) ≤ j % 2 ^ i ⊦ cast(i, j + j0) = cast(i, j0 + j % 2 ^ i - 2 ^ i)

User applied rule insert elim lemma

And obtained the following new goals

Proof step 16:
j = i0 * 2 ^ i + j1 ∧ 0 ≤ j1 ∧ j1 < 2 ^ i ∧ 0 ≤ i0 ∧ 0 < 2 ^ i, 0 ≤ j, 0 ≤ j0, 0 < i, 2 ^ (i - 1) ≤ j1 ⊦ cast(i, j + j0) = cast(i, j0 + j1 - 2 ^ i)

Rule argument

seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [j, 2 ^ i, i0, j1] rotate: T spec: int-div inst: name: div-elim-pospos

Additional information

Used simplifier rules: ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);%⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0); (elim) Used terms: j, 2 ^ i, i0, j1