[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 7 from proof int2int-lastn-negpos (previous step)

bits2int(int2bits(i)) = bits2int(bits + lastn(m, int2bits(i))), # bits = # int2bits(i) - m, int2bits(i) = bits + lastn(m, int2bits(i)), i < 0, m ≤ # int2bits(i) ⊦ bits = [], bits2int(lastn(m, int2bits(i))) = i % 2 ^ n→i(m) + 2 ^ n→i(m), i % 2 ^ n→i(m) = 0, m = 0, lastn(m, int2bits(i)).first

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ # bits0 ≠ 0 ∧ # bits ≠ 0 ∧ bits.first → bits2int(bits + bits0) = ~(2 ^ n→i(# bits0) * bits2uint(bitsinvert(bits))) - bits2uint(bitsinvert(bits0)) + 1
Proof step 9:
¬(# lastn(m, int2bits(i)) ≠ 0 ∧ # bits ≠ 0 ∧ bits.first), bits2int(int2bits(i)) = bits2int(bits + lastn(m, int2bits(i))), # bits = # int2bits(i) - m, int2bits(i) = bits + lastn(m, int2bits(i)), i < 0, m ≤ # int2bits(i) ⊦ bits = [], bits2int(lastn(m, int2bits(i))) = i % 2 ^ n→i(m) + 2 ^ n→i(m), i % 2 ^ n→i(m) = 0, m = 0, lastn(m, int2bits(i)).first
Proof step 11:
# lastn(m, int2bits(i)) ≠ 0 ∧ # bits ≠ 0 ∧ bits.first, bits2int(int2bits(i)) = ~(2 ^ n→i(# lastn(m, int2bits(i))) * bits2uint(bitsinvert(bits))) - bits2uint(bitsinvert(lastn(m, int2bits(i)))) + 1, # bits = # int2bits(i) - m, int2bits(i) = bits + lastn(m, int2bits(i)), i < 0, m ≤ # int2bits(i) ⊦ bits = [], bits2int(lastn(m, int2bits(i))) = i % 2 ^ n→i(m) + 2 ^ n→i(m), i % 2 ^ n→i(m) = 0, m = 0, lastn(m, int2bits(i)).first

Rule argument

name: bits2int-split-01 rotate: F subst: [bits0, bits] → [lastn(m, int2bits(i)), bits]

Additional information

Used terms: lastn(m, int2bits(i)), bits