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Proof step 8 from proof cast-mult-pos (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
0 ≤ j, j % 2 ^ i ≤ 2 ^ (i - 1) - 1, 0 < i ⊦ cast(i, j) = j % 2 ^ i
Proof step 10:
¬(0 ≤ j * 2 ^ i + j0 ∧ (j * 2 ^ i + j0) % 2 ^ i ≤ 2 ^ (i - 1) - 1 ∧ 0 < i), 0 ≤ j, 0 ≤ j0, 0 < i, j0 % 2 ^ i < 2 ^ (i - 1) ⊦ cast(i, j * 2 ^ i + j0) = j0 % 2 ^ i
Proof step 11:
0 ≤ j * 2 ^ i + j0 ∧ (j * 2 ^ i + j0) % 2 ^ i ≤ 2 ^ (i - 1) - 1 ∧ 0 < i, 0 ≤ j, 0 ≤ j0, 0 < i, j0 % 2 ^ i < 2 ^ (i - 1) ⊦ (j * 2 ^ i + j0) % 2 ^ i = j0 % 2 ^ i

Rule argument

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

Additional information

Used terms: j * 2 ^ i + j0, i