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Proof step 61 from proof band-min (previous step)

# int2bits(i3) = # int2bits(i2), ~1 = ((i3 * 2 + j0) + 1) % 2, ~1 = ((i2 * 2 + j1) + 1) % 2, abs(j1) < 2, abs(j0) < 2, i3 * 2 + j0 < ~1, i2 * 2 + j1 < ~1, bits2int(bitsand(int2bits(i3), int2bits(i2))) < i / 2, j0 ≤ 0, i3 ≤ 0, i3 * 2 + j0 < 0, j1 ≤ 0, i2 ≤ 0, i2 * 2 + j1 < 0, 2 ^ (n→i(# int2bits(i2)) + 1) + i < 0, i < 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 ⊦ i→n(i3) < i→n((i3 * 2 + j0) + 1), ~2 = i3 * 2 + j0, ~2 = i2 * 2 + j1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 62:
# int2bits(i3) = # int2bits(i2), ~1 = ((i3 * 2 + j0) + 1) % 2, ~1 = ((i2 * 2 + j1) + 1) % 2, i2 * 2 + j1 < i3 + j0, i3 < 0, ~1 ≤ i3 + j0, abs(j1) < 2, abs(j0) < 2, i3 * 2 + j0 < ~1, i2 * 2 + j1 < ~1, bits2int(bitsand(int2bits(i3), int2bits(i2))) < i / 2, j0 ≤ 0, i3 * 2 + j0 < 0, j1 ≤ 0, i2 ≤ 0, i2 * 2 + j1 < 0, 2 ^ (n→i(# int2bits(i2)) + 1) + i < 0, i < 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 ⊦ i3 = 0, ~2 = i3 * 2 + j0, ~2 = i2 * 2 + j1

Rule argument

Additional information

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