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Proof step 18 from proof band-011x-neg (previous step)

bitsand(false ' + mklist(true, i→n(i)), sgn(bitsinvert(uint2bits(~ j)) +1)) = false ' + bits, j < i, 0 ≤ i, j < 0 ⊦ (j + 1) % 2 ^ i + 2 ^ i = 1 + bits2uint(bits)

User applied rule insert lemma

And obtained the following new goals

Proof step 19:
¬ false ∈ bits0, bits ≠ [], bits.first, bitsand(false ' + bits0, bits) = false ' + bits1 ⊦ bits2uint(bits1) = 2 ^ n→i(# bits0) - bits2uint(bitsinvert(bits)) % 2 ^ n→i(# bits0) + 1
Proof step 20:
bitsand(false ' + mklist(true, i→n(i)), sgn(bitsinvert(uint2bits(~ j)) +1)) = false ' + bits, j < i, 0 ≤ i, j < 0 ⊦ ¬ false ∈ mklist(true, i→n(i)) ∧ sgn(bitsinvert(uint2bits(~ j)) +1) ≠ [] ∧ sgn(bitsinvert(uint2bits(~ j)) +1).first, (j + 1) % 2 ^ i + 2 ^ i = 1 + bits2uint(bits)
Proof step 25:
bits2uint(bits) = 2 ^ n→i(# mklist(true, i→n(i))) - bits2uint(bitsinvert(sgn(bitsinvert(uint2bits(~ j)) +1))) % 2 ^ n→i(# mklist(true, i→n(i))) + 1, bitsand(false ' + mklist(true, i→n(i)), sgn(bitsinvert(uint2bits(~ j)) +1)) = false ' + bits, j < i, 0 ≤ i, j < 0 ⊦ (j + 1) % 2 ^ i + 2 ^ i = 1 + bits2uint(bits)

Rule argument

name: and-011x-neg subst: [bits0, bits, bits1] → [mklist(true, i→n(i)), sgn(bitsinvert(uint2bits(~ j)) +1), bits]

Additional information

Used terms: mklist(true, i→n(i)), sgn(bitsinvert(uint2bits(~ j)) +1), bits