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Proof step 47 from proof bxor-def-logic (previous step)

⊦ bitsand(bitsinvert(int2bits(j)), int2bits(i)) = [], bitsand(sgn(bitsinvert(int2bits(i))), int2bits(j)) = [], bits2int(bitsxor(int2bits(i), int2bits(j))) = bits2int(bitsor(bitsand(bitsinvert(int2bits(j)), int2bits(i)), sgn(bitsand(bitsinvert(int2bits(i)), int2bits(j))))), bitsand(bitsinvert(int2bits(i)), int2bits(j)) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsor(bits, bits0) = bitsor(bits0, bits)
Proof step 49:
bitsand(bitsinvert(int2bits(j)), int2bits(i)) ≠ [] ∧ sgn(bitsand(bitsinvert(int2bits(i)), int2bits(j))) ≠ [] ⊦ bitsand(bitsinvert(int2bits(j)), int2bits(i)) = [], bitsand(sgn(bitsinvert(int2bits(i))), int2bits(j)) = [], bits2int(bitsxor(int2bits(i), int2bits(j))) = bits2int(bitsor(sgn(bitsand(bitsinvert(int2bits(i)), int2bits(j))), bitsand(bitsinvert(int2bits(j)), int2bits(i)))), bitsand(bitsinvert(int2bits(i)), int2bits(j)) = []

Rule argument

name: bitsor-com rotate: F subst: [bits0, bits] → [sgn(bitsand(bitsinvert(int2bits(i)), int2bits(j))), bitsand(bitsinvert(int2bits(j)), int2bits(i))]

Additional information

Used simplifier rules: ⊦ sgn(bits) = [] ↔ bits = []; Used terms: sgn(bitsand(bitsinvert(int2bits(i)), int2bits(j))), bitsand(bitsinvert(int2bits(j)), int2bits(i))