[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 34 from proof uint2bits-restsgn (previous step)

uint2bits(2 * 2 ^ i - i0) = false ' + boolvar0 ' + bits0, 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 ⊦ sgn(boolvar0 ' + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = (boolvar0 ' + bits0) + ((4 * 2 ^ i - i0 * 2) % 2 = 1)', 4 * 2 ^ i = 1 + i0 * 2, (¬ boolvar0) ∈ bits0

User applied rule cut formula

And obtained the following new goals

Proof step 35:
boolvar0, uint2bits(2 * 2 ^ i - i0) = false ' + boolvar0 ' + bits0, 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 ⊦ sgn(boolvar0 ' + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = (boolvar0 ' + bits0) + ((4 * 2 ^ i - i0 * 2) % 2 = 1)', 4 * 2 ^ i = 1 + i0 * 2, (¬ boolvar0) ∈ bits0
Proof step 72:
uint2bits(2 * 2 ^ i - i0) = false ' + boolvar0 ' + bits0, 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 ⊦ boolvar0, sgn(boolvar0 ' + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = (boolvar0 ' + bits0) + ((4 * 2 ^ i - i0 * 2) % 2 = 1)', 4 * 2 ^ i = 1 + i0 * 2, (¬ boolvar0) ∈ bits0

Rule argument

fma: boolvar0

Additional information

Used formulas: boolvar0