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Proof step 5 from proof band-011x-is-unsigned (previous step)

0 ≤ j + 2 ^ (i - 1), j < 0, 0 < i ⊦ band(j, 2 ^ i - 1) = bits2uint(fillsgn(i→n(i), int2bits(j)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ j < 0 ∧ ¬ false ∈ int2bits(i).rest ∧ 0 ≤ i → band(j, i) = (j + 1) % (i + 1) + i
Proof step 7:
j < 0 ∧ ¬ false ∈ int2bits(2 ^ i - 1).rest ∧ 0 ≤ 2 ^ i - 1, 0 ≤ j + 2 ^ (i - 1), j < 0, 0 < i ⊦ (j + 1) % ((2 ^ i - 1) + 1) + (2 ^ i - 1) = bits2uint(fillsgn(i→n(i), int2bits(j)))

Rule argument

name: band-011x-neg-02 rotate: F subst: [i, j] → [2 ^ i - 1, j]

Additional information

Used simplifier rules: band-commutativity: ⊦ band(i, j) = band(j, i);%band-associativity: ⊦ band(i, band(i0, j)) = band(band(i, i0), j);%band-011x-neg: ⊦ j < 0 ∧ 0 ≤ i → band(j, 2 ^ i - 1) = (j + 1) % 2 ^ i + 2 ^ i - 1;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i < j → ¬ j < i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;%0 ≤ i ⊦ int2bits(2 ^ i - 1) = false ' + mklist(true, i→n(i));% ⊦ (boolvar ' + bits).rest = bits;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ ¬ false ∈ mklist(true, n); Used terms: 2 ^ i - 1, j