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Proof step 92 from proof cast-mod-neg-low (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 93:
⊦ abs(i) < j → i % j = i
Proof step 94:
¬ abs(j) < 2 ^ i, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 0, 0 < i, j < 2 ^ (i - 1), 0 ≤ 2 ^ (i - 1) + j, j % 2 ^ i < j ⊦ j = j % 2 ^ i + 2 ^ i

Rule argument

seq: ⊦ abs(i) < j → i % j = i subst: [j, i] → [2 ^ i, j] rotate: F spec: int-div inst: name: mod-zero-05

Additional information

Used simplifier rules: ⊦ i = i + j ↔ j = 0;% ⊦ 0 < j → (i ^ j = 0 ↔ i = 0);% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i < 0 → abs(i) = ~ i;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false); Used terms: 2 ^ i, j