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Proof step 24 from proof intshiftrsign-abs-le (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ i < 0 ∧ 0 < j → (i / j ≤ i ↔ i = 0 ∨ j = 1) subst: [j, i] → [2 ^ band(j, 31), i + 1] rotate: F spec: int-div inst: name: div-le-same-neg

Additional information

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