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Proof step 30 from proof bxor-def-logic (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits)
Proof step 32:
int2bits(i) ≠ [] ∧ sgn(bitsinvert(int2bits(j))) ≠ [] ⊦ bitsand(sgn(bitsinvert(int2bits(i))), int2bits(j)) = [], bits2int(bitsxor(int2bits(i), int2bits(j))) = bits2int(bitsor(sgn(bitsand(sgn(bitsinvert(int2bits(j))), int2bits(i))), sgn(bitsand(bitsinvert(int2bits(i)), int2bits(j))))), bitsand(sgn(bitsinvert(int2bits(j))), int2bits(i)) = []

Rule argument

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

Additional information

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