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Proof step 40 from proof int2bits-add-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 41:
⊦ ¬(∃ i. 2 ^ i - 1 = j) ∧ 0 ≤ j → # int2bits(j) = # int2bits(j + 1)
Proof step 42:
¬(¬(∃ i. 2 ^ i - 1 = ~ j) ∧ 0 ≤ ~ j), 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(~ j)) + j, 0 ≤ (2 ^ n→i(# int2bits(~ j)) + j) * 2 ^ i + j0, 0 ≤ j0 + 2 ^ i, 0 < 2 ^ n→i(# int2bits(1 - j)) + j ⊦ sgn(sgn(uint2bits(2 ^ n→i(# int2bits(~ 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)
Proof step 51:
¬(∃ i. 2 ^ i - 1 = ~ j) ∧ 0 ≤ ~ j, 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(~ j + 1)) + j, 0 ≤ (2 ^ n→i(# int2bits(~ j + 1)) + j) * 2 ^ i + j0, 0 ≤ j0 + 2 ^ i, 0 < 2 ^ n→i(# int2bits(1 - j)) + j ⊦ sgn(sgn(uint2bits(2 ^ n→i(# int2bits(~ j + 1)) + 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)

Rule argument

name: int2bits-len-pos-suc rotate: F subst: [j] → [~ j]

Additional information

Used terms: ~ j