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Proof step 19 from proof uint2bits-add-low (previous step)

j ≤ 2 ^ (i - 1), j ≤ 2 ^ i, 0 ≤ 2 ^ (i - 1), 1 < j + 2 ^ i, j < 2 ^ (i - 1), j + 1 < i, 0 < 2 ^ (i - 1), j < 0, 0 < i, 0 ≤ j + 2 ^ (i - 1), j < i, j + 1 < 2 ^ (i - 1), j ≤ 0, 0 < j + 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ abs(j) ≤ 2 ^ (i0 - 1) → uint2bits(j + 2 ^ i0) = false ' + fillsgn(i→n(i0), int2bits(j)) ⊦ uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (~1 = j % 2)' = false ' + fillsgn(i→n(i), int2bits(j))

User applied rule apply induction

And obtained the following new goals

Proof step 20:
(j - 1) / 2 < 0 ∧ 0 < i - 1 ∧ abs((j - 1) / 2) ≤ 2 ^ (i - 1 - 1) → uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = false ' + fillsgn(i→n(i - 1), int2bits((j - 1) / 2)), j ≤ 2 ^ (i - 1), j ≤ 2 ^ i, 0 ≤ 2 ^ (i - 1), 1 < j + 2 ^ i, j < 2 ^ (i - 1), j + 1 < i, 0 < 2 ^ (i - 1), j < 0, 0 < i, 0 ≤ j + 2 ^ (i - 1), j < i, j + 1 < 2 ^ (i - 1), j ≤ 0, 0 < j + 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ abs(j) ≤ 2 ^ (i0 - 1) → uint2bits(j + 2 ^ i0) = false ' + fillsgn(i→n(i0), int2bits(j)) ⊦ uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (~1 = j % 2)' = false ' + fillsgn(i→n(i), int2bits(j))

Rule argument

subst: [i0, j] → [i - 1, (j - 1) / 2]

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i→n(i - 1) < i→n(i) ↔ 0 < i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: i - 1, (j - 1) / 2