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Proof step 19 from proof cast-law-add-pos (previous step)

0 ≤ i, 0 ≤ i0, 0 ≤ j1, j1 < 2 ^ i, 2 ^ (i - 1) ≤ j1, 0 < i, 0 ≤ j0 ⊦ cast(i, i0 * 2 ^ i + j1 + j0) = cast(i, j0 + j1 - 2 ^ i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ 0 ≤ j ∧ 0 ≤ j0 ∧ 0 < i → cast(i, j * 2 ^ i + j0) = cast(i, j0)
Proof step 21:
0 ≤ i0 ∧ 0 ≤ j1 + j0 ∧ 0 < i, 0 ≤ i, 0 ≤ i0, 0 ≤ j1, j1 < 2 ^ i, 2 ^ (i - 1) ≤ j1, 0 < i, 0 ≤ j0 ⊦ cast(i, j1 + j0) = cast(i, j0 + j1 - 2 ^ i)

Rule argument

name: cast-mult-pos rotate: F subst: [j0, j, i] → [j1 + j0, i0, i]

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ 0 ≤ i ∧ 0 ≤ j → ¬ i + j < 0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ 1 * i = i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i; Used terms: j1 + j0, i0, i