[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 57 from proof bor-2pot-plus-01 (previous step)

0 < i, j0 < 2 ^ i, j < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j + 1 < 2 ^ i, j < j0, j < i, 0 < 2 ^ i, 0 ≤ j0, j + 1 < i, j < 0, # int2bits(j0) ≤ i→n(i) ⊦ bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], i = 0, bits2int(int2bits(j) + bitsinvert(bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i))))) = j * 2 ^ i + j0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 58:
⊦ # bits0 ≠ 0 ∧ # bits ≠ 0 ∧ bits.first → bits2int(bits + bits0) = ~(2 ^ n→i(# bits0) * bits2uint(bitsinvert(bits))) - bits2uint(bitsinvert(bits0)) + 1
Proof step 59:
¬(# bitsinvert(bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i)))) ≠ 0 ∧ # int2bits(j) ≠ 0 ∧ int2bits(j).first), 0 < i, j0 < 2 ^ i, j < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j + 1 < 2 ^ i, j < j0, j < i, 0 < 2 ^ i, 0 ≤ j0, j + 1 < i, j < 0, # int2bits(j0) ≤ i→n(i) ⊦ bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], i = 0, bits2int(int2bits(j) + bitsinvert(bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i))))) = j * 2 ^ i + j0
Proof step 62:
# bitsinvert(bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i)))) ≠ 0 ∧ # int2bits(j) ≠ 0 ∧ int2bits(j).first, 0 < i, j0 < 2 ^ i, j < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j + 1 < 2 ^ i, j < j0, j < i, 0 < 2 ^ i, 0 ≤ j0, j + 1 < i, j < 0, # int2bits(j0) ≤ i→n(i) ⊦ bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = [], i = 0, ~(2 ^ n→i(# bitsinvert(bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i))))) * bits2uint(bitsinvert(int2bits(j)))) - bits2uint(bitsinvert(bitsinvert(bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i)))))) + 1 = j * 2 ^ i + j0

Rule argument

seq: ⊦ # bits0 ≠ 0 ∧ # bits ≠ 0 ∧ bits.first → bits2int(bits + bits0) = ~(2 ^ n→i(# bits0) * bits2uint(bitsinvert(bits))) - bits2uint(bitsinvert(bits0)) + 1 subst: [bits0, bits] → [bitsinvert (bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i)))), int2bits(j)] rotate: F spec: int2bits inst: name: bits2int-split-01

Additional information

Used terms: bitsinvert(bitsand(bitsinvert(int2bits(j0)), mklist(true, i→n(i)))), int2bits(j)