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Proof step 52 from proof int2bits-add-high (previous step)

0 ≤ j0 + 2 ^ i, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0, 0 < 2 ^ n→i(# int2bits(1 - j)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(1 - j)) + j) + j0 ⊦ sgn(sgn(uint2bits(2 ^ n→i(# int2bits(1 - j)) + j - 1) + fillsgn(i→n(i), int2bits(j0).rest)).rest) = sgn(uint2bits(2 ^ n→i(# int2bits(1 - j)) + j - 1).rest) + fillsgn(i→n(i), int2bits(j0).rest), ∃ i0. 1 = 2 ^ i0 + j

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 53:
⊦ 0 < j → sgn(uint2bits(j) + bits) = uint2bits(j) + bits
Proof step 54:
¬ 0 < 2 ^ n→i(# int2bits(1 - j)) + j - 1, 0 ≤ j0 + 2 ^ i, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0, 0 < 2 ^ n→i(# int2bits(1 - j)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(1 - j)) + j) + j0 ⊦ sgn(sgn(uint2bits(2 ^ n→i(# int2bits(1 - j)) + j - 1) + fillsgn(i→n(i), int2bits(j0).rest)).rest) = sgn(uint2bits(2 ^ n→i(# int2bits(1 - j)) + j - 1).rest) + fillsgn(i→n(i), int2bits(j0).rest), ∃ i0. 1 = 2 ^ i0 + j
Proof step 56:
0 < 2 ^ n→i(# int2bits(1 - j)) + j - 1, 0 ≤ j0 + 2 ^ i, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0, 0 < 2 ^ n→i(# int2bits(1 - j)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(1 - j)) + j) + j0 ⊦ sgn((uint2bits(2 ^ n→i(# int2bits(1 - j)) + j - 1) + fillsgn(i→n(i), int2bits(j0).rest)).rest) = sgn(uint2bits(2 ^ n→i(# int2bits(1 - j)) + j - 1).rest) + fillsgn(i→n(i), int2bits(j0).rest), ∃ i0. 1 = 2 ^ i0 + j

Rule argument

name: uint2bits-sgn-02 rotate: F subst: [bits, j] → [fillsgn(i→n(i), int2bits(j0).rest), 2 ^ n→i(# int2bits(1 - j)) + j - 1]

Additional information

Used terms: fillsgn(i→n(i), int2bits(j0).rest), 2 ^ n→i(# int2bits(1 - j)) + j - 1