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Proof step 17 from proof cast-law-mult-pos (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 18:
j1 = i1 * 2 ^ i + j ∧ 0 ≤ j ∧ j < 2 ^ i ∧ 0 ≤ i1 ∧ 0 < 2 ^ i, 0 ≤ j1, 0 < j1, j1 < 2 ^ i, 0 ≤ i, 0 ≤ i0, 0 < i, 2 ^ (i - 1) ≤ j ⊦ cast(i, j1 * j0) = cast(i, (j - 2 ^ i) * j0)

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] → [j1, 2 ^ i, i1, j] 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;%⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0); (elim) Used terms: j1, 2 ^ i, i1, j