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Proof step 44 from proof cast-mod-neg-high (previous step)

j % 2 ^ i = 0, 0 ≤ 2 ^ (i - 1), j + 2 ^ i ≤ 0, j + 2 ^ i < i, i→n(i) ≤ # int2bits(j), 2 ^ (i - 1) + j < 0, 0 < i, j < 0 ⊦ i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j))) = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ j ≤ ~(2 ^ i) ∧ 0 < i ∧ j % 2 ^ i = 0 → int2bits(j) = int2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest
Proof step 46:
j ≤ ~(2 ^ i) ∧ 0 < i ∧ j % 2 ^ i = 0, j % 2 ^ i = 0, 0 ≤ 2 ^ (i - 1), j + 2 ^ i ≤ 0, j + 2 ^ i < i, i→n(i) ≤ #(int2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest), 2 ^ (i - 1) + j < 0, 0 < i, j < 0 ⊦ i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest)) = 0

Rule argument

seq: ⊦ j ≤ ~(2 ^ i) ∧ 0 < i ∧ j % 2 ^ i = 0 → int2bits(j) = int2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest subst: [j, i] → [j, i] rotate: F spec: int2bits inst: name: int2bits-rec-2pot-neg-01

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: j, i