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

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

User applied rule insert elim lemma

And closed the goal!

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: ⊦ i < j → ¬ j < i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i * j = j * i;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0);% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 ≤ j1 ∧ j1 < j → (i0 * j + j1 < j ↔ i0 ≤ 0);% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward)% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i * 0 = 0;% ⊦ i + 0 = 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