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Proof step 39 from proof cast-add-2pot-h (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 40:
0 ≤ j, j % 2 ^ i ≤ 2 ^ (i - 1) - 1, 0 < i ⊦ cast(i, j) = j % 2 ^ i
Proof step 41:
¬(0 ≤ 2 ^ i + j ∧ (2 ^ i + j) % 2 ^ i ≤ 2 ^ (i - 1) - 1 ∧ 0 < i), j < 0, 0 < i, 0 < j + 2 ^ i, 0 < 2 ^ i + j, j + 2 ^ (i - 1) < 0 ⊦ cast(i, j + 2 ^ i) = j + 2 ^ i
Proof step 50:
0 ≤ 2 ^ i + j ∧ (2 ^ i + j) % 2 ^ i ≤ 2 ^ (i - 1) - 1 ∧ 0 < i, j < 0, 0 < i, 0 < j + 2 ^ i, 0 < 2 ^ i + j, j + 2 ^ (i - 1) < 0 ⊦ (2 ^ i + j) % 2 ^ i = j + 2 ^ i

Rule argument

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

Additional information

Used simplifier rules: ⊦ i + j = j + i; Used terms: 2 ^ i + j, i