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Proof step 15 from proof int2int-lastn-negpos (previous step)

# bits = # int2bits(i) - m, int2bits(i) = bits + lastn(m, int2bits(i)), (bits2int(bits) + 1) * 2 ^ n→i(m) = (i + bits2uint(bitsinvert(lastn(m, int2bits(i))))) + 1, bits.first, i < 0, m ≤ # int2bits(i) ⊦ bits2int(lastn(m, int2bits(i))) = i % 2 ^ n→i(m) + 2 ^ n→i(m), i % 2 ^ n→i(m) = 0, m = 0, bits = [], lastn(m, int2bits(i)).first

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 16:
⊦ bits ≠ [] ∧ ¬ bits.first → bits2uint(bitsinvert(bits)) = 2 ^ n→i(# bits) + ~ bits2int(bits) - 1
Proof step 17:
lastn(m, int2bits(i)) ≠ [] ∧ ¬ lastn(m, int2bits(i)).first, # bits = # int2bits(i) - m, int2bits(i) = bits + lastn(m, int2bits(i)), (bits2int(bits) + 1) * 2 ^ n→i(m) = (i + (2 ^ n→i(# lastn(m, int2bits(i))) + ~ bits2int(lastn(m, int2bits(i))) - 1)) + 1, bits.first, i < 0, m ≤ # int2bits(i) ⊦ bits2int(lastn(m, int2bits(i))) = i % 2 ^ n→i(m) + 2 ^ n→i(m), i % 2 ^ n→i(m) = 0, m = 0, bits = [], lastn(m, int2bits(i)).first

Rule argument

name: bits2uint-inv-neg rotate: F subst: [bits] → [lastn(m, int2bits(i))]

Additional information

Used simplifier rules: ⊦ lastn(n, bits) = [] ↔ n = 0; Used terms: lastn(m, int2bits(i))