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Proof step 8 from proof cast-if (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
0 ≤ j, 2 ^ (i - 1) - 1 < j % 2 ^ i, 0 < i ⊦ cast(i, j) = j % 2 ^ i - 2 ^ i

Rule argument

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

Additional information

Used simplifier rules: ⊦ i + (j - i) = j;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: j, i