[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 10 from proof uint2bits-add-high (previous step)

(1 - 2 ^ i) + 2 ^ i = 1, (2 ^ (i - 1) + (1 - 2 ^ i)) + 1 < 2 ^ i, 1 - 2 ^ i < 2 ^ i, (1 - 2 ^ i) + 1 < 2 ^ i, 1 - 2 ^ i < i, 2 ^ (i - 1) + (1 - 2 ^ i) < 0, 0 < i, 1 - 2 ^ i < 0, 2 ^ (i - 1) < 2 ^ i, (2 ^ (i - 1) + (1 - 2 ^ i)) + 1 < i, 2 ^ (i - 1) + (1 - 2 ^ i) < i, 0 < 2 ^ i, (1 - 2 ^ i) + 1 < i, 2 ^ (i - 1) + (1 - 2 ^ i) < 2 ^ i, 2 ^ (i - 1) + (1 - 2 ^ i) ≤ 0, 1 - 2 ^ i ≤ 0, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ false ' + true ' = sgn(int2bits(1 - 2 ^ i).rest)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ j < 0 → int2bits(j) = sgn(int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest)
Proof step 12:
1 - 2 ^ i < 0, (1 - 2 ^ i) + 2 ^ i = 1, (2 ^ (i - 1) + (1 - 2 ^ i)) + 1 < 2 ^ i, 1 - 2 ^ i < 2 ^ i, (1 - 2 ^ i) + 1 < 2 ^ i, 1 - 2 ^ i < i, 2 ^ (i - 1) + (1 - 2 ^ i) < 0, 0 < i, 1 - 2 ^ i < 0, 2 ^ (i - 1) < 2 ^ i, (2 ^ (i - 1) + (1 - 2 ^ i)) + 1 < i, 2 ^ (i - 1) + (1 - 2 ^ i) < i, 0 < 2 ^ i, (1 - 2 ^ i) + 1 < i, 2 ^ (i - 1) + (1 - 2 ^ i) < 2 ^ i, 2 ^ (i - 1) + (1 - 2 ^ i) ≤ 0, 1 - 2 ^ i ≤ 0, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ false ' + true ' = sgn(sgn(int2bits(2 ^ n→i(# int2bits(~(1 - 2 ^ i))) + (1 - 2 ^ i)).rest).rest)

Rule argument

name: int2bits-neg-03 rotate: F subst: [j] → [1 - 2 ^ i]

Additional information

Used terms: 1 - 2 ^ i