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Proof step 81 from proof cast-mod-neg-low (previous step)

i→n(i) = # int2bits(j % 2 ^ i).rest, i→n(i) < # int2bits(j % 2 ^ i), 0 < i, j < 0, j % 2 ^ i + 2 ^ (i - 1) < 0, j < 2 ^ (i - 1), 2 ^ (i - 1) + j < 0, j + 2 ^ i ≤ 0, j + 2 ^ i < i, j + 2 ^ i < j % 2 ^ i, 0 < j % 2 ^ i ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(int2bits(j % 2 ^ i).rest) = j % 2 ^ i + 2 ^ i, ~1 = j % 2 ^ i

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 82:
⊦ i < 0 ∧ j ≠ 0 → ¬ 0 < i % j

Rule argument

seq: ⊦ i < 0 ∧ j ≠ 0 → ¬ 0 < i % j subst: [j, i] → [2 ^ i, j] rotate: F spec: int-div inst: name: mod-grz-01

Additional information

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