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Proof step 22 from proof uint2bits-restsgn (previous step)

i0 * 2 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 2 ^ i < i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ i0, 0 < i0, 1 < i, 1 < i0, 0 < i, 0 ≤ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 4 * 2 ^ i = 1 + i0 * 2, sgn(uint2bits(2 * 2 ^ i - i0).rest + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = uint2bits(2 * 2 ^ i - i0).rest + ((4 * 2 ^ i - i0 * 2) % 2 = 1)'

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
(¬ bits.first) ∈ bits.rest, bits ≠ [] ⊦ sgn(bits + boolvar0 ') = sgn(bits) + boolvar0 '
Proof step 24:
¬((¬ uint2bits(2 * 2 ^ i - i0).rest.first) ∈ uint2bits(2 * 2 ^ i - i0).rest.rest ∧ uint2bits(2 * 2 ^ i - i0).rest ≠ []), i0 * 2 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 2 ^ i < i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ i0, 0 < i0, 1 < i, 1 < i0, 0 < i, 0 ≤ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 4 * 2 ^ i = 1 + i0 * 2, sgn(uint2bits(2 * 2 ^ i - i0).rest + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = uint2bits(2 * 2 ^ i - i0).rest + ((4 * 2 ^ i - i0 * 2) % 2 = 1)'
Proof step 75:
(¬ uint2bits(2 * 2 ^ i - i0).rest.first) ∈ uint2bits(2 * 2 ^ i - i0).rest.rest ∧ uint2bits(2 * 2 ^ i - i0).rest ≠ [], i0 * 2 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 2 ^ i < i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ i0, 0 < i0, 1 < i, 1 < i0, 0 < i, 0 ≤ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 4 * 2 ^ i = 1 + i0 * 2, sgn(uint2bits(2 * 2 ^ i - i0).rest) + ((4 * 2 ^ i - i0 * 2) % 2 = 1)' = uint2bits(2 * 2 ^ i - i0).rest + ((4 * 2 ^ i - i0 * 2) % 2 = 1)'

Rule argument

seq: (¬ bits.first) ∈ bits.rest, bits ≠ [] ⊦ sgn(bits + boolvar0 ') = sgn(bits) + boolvar0 ' subst: [boolvar0, bits] → [(4 * 2 ^ i - i0 * 2) % 2 = 1, uint2bits(2 * 2 ^ i - i0).rest] rotate: F spec: bitopsaux inst: name: sgn-last-03

Additional information

Used terms: (4 * 2 ^ i - i0 * 2) % 2 = 1, uint2bits(2 * 2 ^ i - i0).rest