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

bits2int(sgn(bitsinvert(uint2bits(~(j / 2 ^ i))))) = bits2int(sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1)), j % 2 ^ i < 0, true ∈ uint2bits(~(j / 2 ^ i)), false ∈ uint2bits(~(j / 2 ^ i)), true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest, 0 ≤ i, j < 0, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 0 < i, j / 2 ^ i ≤ 0, j / 2 ^ i < i ⊦ sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) + bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1 = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1) + fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, j % 2 ^ i = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 66:
⊦ bits ≠ [] ∧ bits0 ≠ [] → (bits2int(bits) = bits2int(bits0) ↔ sgn(bits) = sgn(bits0))
Proof step 67:
sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) ≠ [] ∧ sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1) ≠ [], sgn(sgn(bitsinvert(uint2bits(~(j / 2 ^ i))))) = sgn(sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1)), j % 2 ^ i < 0, true ∈ uint2bits(~(j / 2 ^ i)), false ∈ uint2bits(~(j / 2 ^ i)), true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest, 0 ≤ i, j < 0, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 0 < i, j / 2 ^ i ≤ 0, j / 2 ^ i < i ⊦ sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) + bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1 = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1) + fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, j % 2 ^ i = 0

Rule argument

name: bits2int-inj rotate: F subst: [bits0, bits] → [sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1), sgn(bitsinvert(uint2bits(~(j / 2 ^ i))))]

Additional information

Used simplifier rules: bits2int-sgn: ⊦ bits2int(sgn(bits)) = bits2int(bits);%uint2bits-empty: 0 ≤ i ⊦ uint2bits(i) ≠ [];% ⊦ sgn(bits) = [] ↔ bits = [];% ⊦ bitsinvert(bits) = [] ↔ bits = [];% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + 0 = i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ ¬ bits +1 ≠ [] ↔ false; Used terms: sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1), sgn(bitsinvert(uint2bits(~(j / 2 ^ i))))