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Proof step 84 from proof band-min (previous step)

# int2bits(i3) = # int2bits(i / 2), j0 ≤ 0, ~1 ≤ i3 + j0, i0 < 0, bits2int(bitsand(int2bits(i3), int2bits(i / 2))) < i0 / 2, i < 0, i < ~1, i3 * 2 + j0 < ~1, abs(j0) < 2, 2 ^ (n→i(# int2bits(i3)) + 1) + i0 ≤ 0, i < i3 + j0, i3 * 2 + j0 < 0, i3 < 0, ∀ i0, j, i1. i→n(i1) < i→n((i3 * 2 + j0) + 1) → # int2bits(i1) = # int2bits(j) ∧ bits2int(bitsand(int2bits(i1), int2bits(j))) < i0 ∧ i1 ≤ 0 ∧ j ≤ 0 ∧ i0 + 2 ^ n→i(# int2bits(j)) ≤ 0 → false ⊦ ~2 = i, ~2 = i3 * 2 + j0, ~1 = ((i3 * 2 + j0) + 1) % 2 ∧ ~1 = (i + 1) % 2

User applied rule insert elim lemma

And obtained the following new goals

Proof step 85:
i = i2 * 2 + j1 ∧ j1 ≤ 0 ∧ abs(j1) < 2 ∧ i2 ≤ 0 ∧ 0 < 2, # int2bits(i3) = # int2bits(i2), j0 ≤ 0, ~1 ≤ i3 + j0, i0 < 0, bits2int(bitsand(int2bits(i3), int2bits(i2))) < i0 / 2, i < 0, i < ~1, i3 * 2 + j0 < ~1, abs(j0) < 2, 2 ^ (n→i(# int2bits(i3)) + 1) + i0 ≤ 0, i < i3 + j0, i3 * 2 + j0 < 0, i3 < 0, ∀ i0, j, i1. i→n(i1) < i→n((i3 * 2 + j0) + 1) → # int2bits(i1) = # int2bits(j) ∧ bits2int(bitsand(int2bits(i1), int2bits(j))) < i0 ∧ i1 ≤ 0 ∧ j ≤ 0 ∧ i0 + 2 ^ n→i(# int2bits(j)) ≤ 0 → false ⊦ ~2 = i, ~2 = i3 * 2 + j0, ~1 = ((i3 * 2 + j0) + 1) % 2 ∧ ~1 = (i + 1) % 2

Rule argument

seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0) subst: [i, j, i0, j0] → [i, 2, i2, j1] rotate: T spec: int-div inst: name: div-elim-negpos

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 < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0); (elim) Used terms: i, 2, i2, j1