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Proof step 68 from proof cast-mod-neg-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 69:
⊦ 0 < j ∧ i < 0 → ¬ 0 < i % j

Rule argument

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

Additional information

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