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Proof step 83 from proof band-min (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 84:
# 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

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j;% ⊦ 0 < ~ i ↔ i < 0;% ⊦ i * ~1 = ~ i;% ⊦ (* i1) * j < j ↔ 0 < (1 - i1) * j;% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 1 * i = i;% ⊦ j ≤ (* i1) * j + i ↔ 0 ≤ (i1 - 1) * j + i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;