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Proof step 3 from proof int2bits-rec-2pot-neg-03 (previous step)

¬(j ≤ ~(2 ^ i) ∧ 0 < i ∧ j % 2 ^ i ≠ 0) ⊦ j ≤ ~(2 ^ i) ∧ 0 < i → int2bits(j) = int2bits((j - (2 ^ i - 1)) / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ 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 5:
j ≤ ~(2 ^ i) ∧ 0 < i ∧ j % 2 ^ i = 0, ¬(j ≤ ~(2 ^ i) ∧ 0 < i ∧ j % 2 ^ i ≠ 0) ⊦ j ≤ ~(2 ^ i) ∧ 0 < i → int2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest = int2bits((j - (2 ^ i - 1)) / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest

Rule argument

name: int2bits-rec-2pot-neg-01 rotate: F subst: [j, i] → [j, i]

Additional information

Used simplifier rules: ⊦ i - (j - j0) = i + j0 - j;% ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1;% ⊦ 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