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Proof step 21 from proof bor-2pot-plus-01 (previous step)

j < 2 ^ i, j0 < 2 ^ i, 0 ≤ i, 0 ≤ j0, 0 < 2 ^ i, j < i, j < j0, j + 1 < 2 ^ i, j < 0 ⊦ i = 0, bits2int(bitsinvert(bitsand(bitsinvert(sgn(bitsinvert(uint2bits(~ j)) +1 + mklist(false, i→n(i)))), bitsinvert(int2bits(j0))))) = j * 2 ^ i + j0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ bits0 ≠ [] ∧ (¬ bits0.first) ∈ bits → sgn(bits + bits0) = sgn(bits) + bits0
Proof step 23:
¬(mklist(false, i→n(i)) ≠ [] ∧ (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1), j < 2 ^ i, j0 < 2 ^ i, 0 ≤ i, 0 ≤ j0, 0 < 2 ^ i, j < i, j < j0, j + 1 < 2 ^ i, j < 0 ⊦ i = 0, bits2int(bitsinvert(bitsand(bitsinvert(sgn(bitsinvert(uint2bits(~ j)) +1 + mklist(false, i→n(i)))), bitsinvert(int2bits(j0))))) = j * 2 ^ i + j0
Proof step 26:
mklist(false, i→n(i)) ≠ [] ∧ (¬ mklist(false, i→n(i)).first) ∈ bitsinvert(uint2bits(~ j)) +1, j < 2 ^ i, j0 < 2 ^ i, 0 ≤ i, 0 ≤ j0, 0 < 2 ^ i, j < i, j < j0, j + 1 < 2 ^ i, j < 0 ⊦ i = 0, bits2int(bitsinvert(bitsand(bitsinvert(sgn(bitsinvert(uint2bits(~ j)) +1) + mklist(false, i→n(i))), bitsinvert(int2bits(j0))))) = j * 2 ^ i + j0

Rule argument

seq: ⊦ bits0 ≠ [] ∧ (¬ bits0.first) ∈ bits → sgn(bits + bits0) = sgn(bits) + bits0 subst: [bits0, bits] → [mklist(false, i→n(i)), bitsinvert(uint2bits(~ j)) +1] rotate: F spec: bitopsaux inst: name: sgn-append-06

Additional information

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