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Proof step 58 from proof int2bits-add-low (previous step)

0 ≤ j0 + 2 ^ (i - 1), 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, j * 2 ^ i + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, 0 < 2 ^ n→i(# int2bits(~(j - 1) - 1)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~(j - 1) - 1)) + j) + j0 ⊦ j * 2 ^ i + j0 = 0, sgn(uint2bits(2 ^ n→i(# int2bits(~(j - 1) - 1)) + j - 1).rest + fillsgn(i→n(i), int2bits(j0))) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 59:
⊦ 0 ≤ i → int2bits(i) = uint2bits(i)
Proof step 60:
0 ≤ 2 ^ n→i(# int2bits(~(j - 1) - 1)) + j - 1, 0 ≤ j0 + 2 ^ (i - 1), 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, j * 2 ^ i + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, 0 < 2 ^ n→i(# int2bits(~(j - 1) - 1)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~(j - 1) - 1)) + j) + j0 ⊦ j * 2 ^ i + j0 = 0, sgn(int2bits(2 ^ n→i(# int2bits(~(j - 1) - 1)) + j - 1).rest + fillsgn(i→n(i), int2bits(j0))) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

name: int2bits-pos rotate: T subst: [i] → [2 ^ n→i(# int2bits(~(j - 1) - 1)) + j - 1]

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ~(i - 1) = ~ i + 1;% ⊦ i + ~ j = i - j;% ⊦ i - j - i0 = i - j + i0;% ⊦ i - j + i = ~ j;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: 2 ^ n→i(# int2bits(~(j - 1) - 1)) + j - 1