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Proof step 74 from proof band-max (previous step)

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

User applied rule apply induction

And closed the goal!

Rule argument

subst: [i0, j] → [i / 2, i1 / 2]

Additional information

Used simplifier rules: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j;% ⊦ 0 ≤ i / 2 ↔ ~1 ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j);%j ≠ 0 ⊦ abs(i / j) = abs(i) / abs(j);% ⊦ 1 < i → abs(i) = i;% ⊦ 0 < j ∧ 0 ≤ i → (i / (* j) < j0 ↔ i < j * j0);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < i → (i * j ≤ i ↔ j ≤ 1);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i; Used terms: i / 2, i1 / 2