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Proof step 39 from proof bor-2pot-plus-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 40:
⊦ ¬ false ∈ bits ∧ # bits ≤ # bits0 ∧ bits ≠ [] → bitsand(bits0, bits) = bits0
Proof step 41:
¬(¬ false ∈ mklist(true, i→n(i)) ∧ # mklist(true, i→n(i)) ≤ # bitsinvert(int2bits(j0).rest) ∧ mklist(true, i→n(i)) ≠ []), # int2bits(j0) = i→n(i) + 1, j < 0, j + 1 < i, 0 ≤ j0, 0 < 2 ^ i, j < i, j < j0, j + 1 < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j < 2 ^ i, j0 < 2 ^ i, 0 < i ⊦ bits2int(int2bits(j) + bitsinvert(bitsand(bitsinvert(int2bits(j0).rest), mklist(true, i→n(i))))) = j * 2 ^ i + j0, i = 0, bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = []
Proof step 44:
¬ false ∈ mklist(true, i→n(i)) ∧ # mklist(true, i→n(i)) ≤ # bitsinvert(int2bits(j0).rest) ∧ mklist(true, i→n(i)) ≠ [], # int2bits(j0) = i→n(i) + 1, j < 0, j + 1 < i, 0 ≤ j0, 0 < 2 ^ i, j < i, j < j0, j + 1 < 2 ^ i, (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j < 2 ^ i, j0 < 2 ^ i, 0 < i ⊦ bits2int(int2bits(j) + bitsinvert(bitsinvert(int2bits(j0).rest))) = j * 2 ^ i + j0, i = 0, bitsinvert(int2bits(j)) + mklist(true, i→n(i)) = []

Rule argument

name: add-ones-lelen-01 rotate: F subst: [bits, bits0] → [mklist(true, i→n(i)), bitsinvert(int2bits(j0).rest)]

Additional information

Used terms: mklist(true, i→n(i)), bitsinvert(int2bits(j0).rest)