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Proof step 66 from proof int2bits-add-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 67:
⊦ 0 < j → (false ∈ uint2bits(j).rest ↔ ¬(∃ i. j = 2 ^ i - 1))
Proof step 68:
0 < 2 ^ n→i(# int2bits(1 - j)) + j - 1, 0 < 2 ^ n→i(# int2bits(1 - j)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(1 - j)) + j) + j0, j * 2 ^ i + j0 < 0, 0 ≤ i, 0 < 2 ^ i + j0, 2 ^ (i - 1) + j0 < 0, 0 < i, j0 < 0, j < 0, 0 ≤ j0 + 2 ^ i, 1 < 2 ^ n→i(# int2bits(1 - j)) + j ⊦ sgn(uint2bits(2 ^ n→i(# int2bits(1 - j)) + j - 1).rest + fillsgn(i→n(i), int2bits(j0).rest)) = sgn(uint2bits(2 ^ n→i(# int2bits(1 - j)) + j - 1).rest) + fillsgn(i→n(i), int2bits(j0).rest), ∀ i. 2 ^ n→i(# int2bits(1 - j)) + j - 1 ≠ 2 ^ i - 1, ∃ i0. 1 = 2 ^ i0 + j

Rule argument

name: uint2bits-falsein-01 rotate: F subst: [j] → [2 ^ n→i(# int2bits(1 - j)) + j - 1]

Additional information

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