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Proof step 3 from proof uint2bits-add-low (previous step)

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), ∀ 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(j + 2 ^ i) = false ' + fillsgn(i→n(i), int2bits(j))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
1 < i ⊦ uint2bits((* i)) = uint2bits(i / 2) + (i % 2 = 1)
Proof step 5:
¬ 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), ∀ 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(j + 2 ^ i) = false ' + fillsgn(i→n(i), int2bits(j))
Proof step 12:
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), ∀ 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((j + 2 ^ i) / 2) + ((j + 2 ^ i) % 2 = 1) = false ' + fillsgn(i→n(i), int2bits(j))

Rule argument

name: uint2bits-pos-num rotate: F subst: [i] → [j + 2 ^ i]

Additional information

Used terms: j + 2 ^ i