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Proof step 99 from proof band-min (previous step)

i0 = i * 2 + j2 ∧ j2 ≤ 0 ∧ abs(j2) < 2 ∧ i ≤ 0 ∧ 0 < 2, # int2bits(i3) = # int2bits(i2), abs(j1) < 2, abs(j0) < 2, i3 * 2 + j0 < ~1, i2 * 2 + j1 < ~1, bits2int(bitsand(int2bits(i3), int2bits(i2))) < i, i0 < 0, j0 ≤ 0, i3 ≤ 0, i3 * 2 + j0 < 0, j1 ≤ 0, j1 < i + 2 ^ n→i(# int2bits(i2)), i2 ≤ 0, i2 < i + 2 ^ n→i(# int2bits(i2)), i2 * 2 + j1 < 0, j0 < i + 2 ^ n→i(# int2bits(i2)), i3 < i + 2 ^ n→i(# int2bits(i2)), 2 ^ (n→i(# int2bits(i2)) + 1) + i0 ≤ 0, 2 ^ (n→i(# int2bits(i2)) + 1) + i0 < i + 2 ^ n→i(# int2bits(i2)), 0 < i + 2 ^ n→i(# int2bits(i2)), ∀ 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 = i3 * 2 + j0, ~2 = i2 * 2 + j1, ~1 = ((i3 * 2 + j0) + 1) % 2 ∧ ~1 = ((i2 * 2 + j1) + 1) % 2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 100:
# int2bits(i3) = # int2bits(i2), 2 ^ (n→i(# int2bits(i2)) + 1) + i * 2 + j2 ≤ 0, i2 < i + 2 ^ n→i(# int2bits(i2)), j1 < i + 2 ^ n→i(# int2bits(i2)), i3 < i + 2 ^ n→i(# int2bits(i2)), j0 < i + 2 ^ n→i(# int2bits(i2)), i * 2 + j2 < 0, bits2int(bitsand(int2bits(i3), int2bits(i2))) < 0, i ≤ 0, j2 < i + 2 ^ n→i(# int2bits(i2)), j2 ≤ 0, 0 < i + 2 ^ n→i(# int2bits(i2)), i2 * 2 + j1 < 0, i2 ≤ 0, j1 ≤ 0, i3 * 2 + j0 < 0, i3 ≤ 0, j0 ≤ 0, bits2int(bitsand(int2bits(i3), int2bits(i2))) < i, i2 * 2 + j1 < ~1, i3 * 2 + j0 < ~1, abs(j0) < 2, abs(j1) < 2, abs(j2) < 2, i + 2 ^ (n→i(# int2bits(i2)) + 1) + j2 < 2 ^ n→i(# int2bits(i2)), ∀ 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 = i2 * 2 + j1, ~2 = i3 * 2 + j0, ~1 = ((i3 * 2 + j0) + 1) % 2 ∧ ~1 = ((i2 * 2 + j1) + 1) % 2

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ n→i(n);% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ i < i + j ↔ 0 < j;% ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 1 * i = i;% ⊦ (* i2) * i + j < i + j0 ↔ (i2 - 1) * i + j < j0;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;