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Proof step 47 from proof int2bits-add-low (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ j < 0 ∧ j0 < 0 ∧ abs(j0) < 2 ^ i ∧ 0 ≤ i → # int2bits(~(j * 2 ^ i + j0)) = # int2bits(~ j) + i→n(i)
Proof step 49:
¬(j < 0 ∧ j0 < 0 ∧ abs(j0) < 2 ^ i ∧ 0 ≤ i), 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, i < n→i(# int2bits(~(j * 2 ^ i + j0))), j * 2 ^ i + j0 < 0, 0 < j + 2 ^ (n→i(# int2bits(~(j * 2 ^ i + j0))) - i), 0 ≤ (j + 2 ^ (n→i(# int2bits(~(j * 2 ^ i + j0))) - i)) * 2 ^ i + j0, 0 ≤ j0 + 2 ^ (i - 1) ⊦ sgn(uint2bits(j + 2 ^ (n→i(# int2bits(~(j * 2 ^ i + j0))) - i) - 1).rest + fillsgn(i→n(i), int2bits(j0))) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0)), j * 2 ^ i + j0 = 0
Proof step 55:
j < 0 ∧ j0 < 0 ∧ abs(j0) < 2 ^ i ∧ 0 ≤ i, 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, i < n→i(# int2bits(~ j) + i→n(i)), j * 2 ^ i + j0 < 0, 0 < j + 2 ^ (n→i(# int2bits(~ j) + i→n(i)) - i), 0 ≤ (j + 2 ^ (n→i(# int2bits(~ j) + i→n(i)) - i)) * 2 ^ i + j0, 0 ≤ j0 + 2 ^ (i - 1) ⊦ sgn(uint2bits(j + 2 ^ (n→i(# int2bits(~ j) + i→n(i)) - i) - 1).rest + fillsgn(i→n(i), int2bits(j0))) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0)), j * 2 ^ i + j0 = 0

Rule argument

name: int2bits-len-add rotate: F subst: [j0, i, j] → [j0, i, j]

Additional information

Used terms: j0, i, j