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Proof step 14 from proof cast-mult-neg (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
j < 0, ~(2 ^ (i - 1)) ≤ j % 2 ^ i, 0 < i ⊦ cast(i, j) = j % 2 ^ i
Proof step 16:
¬(j * 2 ^ i + j0 < 0 ∧ ~(2 ^ (i - 1)) ≤ (j * 2 ^ i + j0) % 2 ^ i ∧ 0 < i), 0 ≤ 2 ^ (i - 1) + j0 % 2 ^ i, 0 < i, j0 < 0, j < 0 ⊦ cast(i, j * 2 ^ i + j0) = j0 % 2 ^ i

Rule argument

name: cast-mod-neg-high rotate: F subst: [j, i] → [j * 2 ^ i + j0, i]

Additional information

Used simplifier rules: ⊦ j < 0 ∧ j0 < 0 ∧ 0 < i → (j * 2 ^ i + j0) % 2 ^ i = j0 % 2 ^ i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i; Used terms: j * 2 ^ i + j0, i