[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 100 from proof int2bits-rec-2pot-neg-02 (previous step)

sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1), i→n(i) + 1 < # bitsinvert(uint2bits(~(j % 2 ^ i))) +1, true ∈ uint2bits(~(j % 2 ^ i)).rest, false ∈ fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i)))).rest, j / 2 ^ i < i, j / 2 ^ i ≤ 0, 0 < i, j + 2 ^ i ≤ 0, j + 2 ^ i < i, j < 0, 0 ≤ i, true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest, false ∈ uint2bits(~(j / 2 ^ i)), true ∈ uint2bits(~(j / 2 ^ i)), j % 2 ^ i < 0 ⊦ bitsinvert(uint2bits(1 - j / 2 ^ i)) +1 = [], j % 2 ^ i = 0, fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i))) +1).rest = fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))) = [], fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i)))) = [], bitsinvert(uint2bits(~(j % 2 ^ i))) +1 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ false ∈ bits → # bits +1 = # bits subst: [bits] → [bitsinvert(uint2bits(~(j % 2 ^ i)))] rotate: F spec: bitopsaux inst: name: succ-len

Additional information

Used simplifier rules: uint2bits-onein-01: ⊦ j < 0 → true ∈ uint2bits(~ j);% ⊦ ¬ false ∈ bits → bits +1 = true + bitsinvert(bits);% ⊦ bitsinvert(bitsinvert(bits)) = bits;% ⊦ boolvar + bits = boolvar ' + bits;% ⊦ boolvar ' + bits ≠ [];% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ m + n < m + n0 ↔ n < n0;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ false ∈ bitsinvert(bits) ↔ true ∈ bits; Used terms: bitsinvert(uint2bits(~(j % 2 ^ i)))