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Proof step 20 from proof intshiftrsign-abs-le (previous step)

i + 1 < 2 ^ band(j, 31), band(j, 31) < 32, i < 0, 0 < band(j, 31), 0 < (2 ^ band(j, 31) + ~(i + 1)) / 2 ^ band(j, 31) + i ⊦ band(j, 31) = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [2 ^ band(j, 31), ~(i + 1), 2 ^ band(j, 31)] rotate: F spec: int-div inst: name: div-add1-pospos

Additional information

Used simplifier rules: band-commutativity: ⊦ band(i, j) = band(j, i);%band-notneg: ⊦ 0 ≤ i → ¬ band(i, j) < 0;%band-associativity: ⊦ band(i, band(i0, j)) = band(band(i, i0), j);% ⊦ i + ~ j = i - j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 < j → (i ^ j = 0 ↔ i = 0);%i ≠ 0 ⊦ i % i = 0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ ¬ 2 ^ i < 0;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j;% ⊦ i ≤ ~1 ↔ i < 0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ 0 < 2 ^ i ↔ 0 ≤ i; Used terms: 2 ^ band(j, 31), ~(i + 1), 2 ^ band(j, 31)