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Proof step 81 from proof band-min (previous step)

# int2bits(i2 / 2) = # int2bits(i / 2), i2 < ~1, i2 < 0, i < ~1, 2 ^ (n→i(# int2bits(i / 2)) + 1) + i0 ≤ 0, i < 0, bits2int(bitsand(int2bits(i2 / 2), int2bits(i / 2))) < i0 / 2, i0 < 0, ∀ 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 ⊦ i→n(i2 / 2) < i→n(i2 + 1), ~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 82:
# int2bits(i2 / 2) = # int2bits(i / 2), i2 < ~1, i2 < 0, i < ~1, 2 ^ (n→i(# int2bits(i / 2)) + 1) + i0 ≤ 0, i < 0, bits2int(bitsand(int2bits(i2 / 2), int2bits(i / 2))) < i0 / 2, i0 < 0, i2 / 2 ≤ 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

Rule argument

Additional information

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