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Proof step 11 from proof cast-add-2pot-h (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
0 ≤ j, 2 ^ (i - 1) - 1 < j % 2 ^ i, 0 < i ⊦ cast(i, j) = j % 2 ^ i - 2 ^ i
Proof step 13:
¬(0 ≤ j + 2 ^ i ∧ 2 ^ (i - 1) - 1 < (j + 2 ^ i) % 2 ^ i ∧ 0 < i), 0 ≤ j + 2 ^ (i - 1), 0 < 2 ^ i + j, j < 0, 0 < i, 0 < j + 2 ^ i ⊦ cast(i, j + 2 ^ i) = j
Proof step 24:
0 ≤ j + 2 ^ i ∧ 2 ^ (i - 1) - 1 < (j + 2 ^ i) % 2 ^ i ∧ 0 < i, 0 ≤ j + 2 ^ (i - 1), 0 < 2 ^ i + j, j < 0, 0 < i, 0 < j + 2 ^ i ⊦ (j + 2 ^ i) % 2 ^ i - 2 ^ i = j

Rule argument

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

Additional information

Used terms: j + 2 ^ i, i