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Proof step 118 from proof cast-mod-neg-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 119:
⊦ 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 → abs(i) = ~ i;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;% ⊦ i + j ≤ i0 + j ↔ i ≤ i0;% ⊦ 0 < i → ¬ 2 ^ i ≤ 2 ^ (i - 1);% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i; Used terms: 2 ^ i, j