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Proof step 54 from proof cast-add-2pot-h (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 55:
⊦ abs(i) < j → i % j = 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 ≤ 0 ∧ 0 ≤ j → i ≤ j;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;%i ≠ 0 ⊦ i % i = 0;% ⊦ 0 ≤ j → 2 ^ j ≠ 0;% ⊦ i + 0 = i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i < 0 → abs(i) = ~ i;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) Used terms: 2 ^ i, j