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Proof step 25 from proof int2bits-add-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 26:
⊦ j0 < 0 ∧ 0 < i ∧ 2 ^ (i - 1) < abs(j0) ∧ abs(j0) ≤ 2 ^ i ∧ 0 < j → uint2bits(j * 2 ^ i + j0) = sgn(uint2bits(j - 1) + fillsgn(i→n(i), int2bits(j0).rest))
Proof step 27:
¬(j0 < 0 ∧ 0 < i ∧ 2 ^ (i - 1) < abs(j0) ∧ abs(j0) ≤ 2 ^ i ∧ 0 < 2 ^ n→i(# int2bits(~ j)) + j), 0 ≤ (2 ^ n→i(# int2bits(~ j)) + j) * 2 ^ i + j0, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0 ⊦ sgn(uint2bits((2 ^ n→i(# int2bits(~ j)) + j) * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0).rest)
Proof step 28:
j0 < 0 ∧ 0 < i ∧ 2 ^ (i - 1) < abs(j0) ∧ abs(j0) ≤ 2 ^ i ∧ 0 < 2 ^ n→i(# int2bits(~ j)) + j, 0 ≤ (2 ^ n→i(# int2bits(~ j)) + j) * 2 ^ i + j0, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0 ⊦ sgn(sgn(uint2bits(2 ^ n→i(# int2bits(~ j)) + j - 1) + fillsgn(i→n(i), int2bits(j0).rest)).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0).rest)

Rule argument

name: uint2bits-add-high-02 rotate: F subst: [j0, i, j] → [j0, i, 2 ^ n→i(# int2bits(~ j)) + j]

Additional information

Used terms: j0, i, 2 ^ n→i(# int2bits(~ j)) + j