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Proof step 16 from proof band-associativity (previous step)

⊦ bitsand(int2bits(i), sgn(bitsand(int2bits(i0), int2bits(j)))) = [], bitsand(sgn(bitsand(int2bits(i), int2bits(i0))), int2bits(j)) = [], sgn(bitsand(int2bits(i), sgn(bitsand(int2bits(i0), int2bits(j))))) = sgn(bitsand(bitsand(int2bits(i), int2bits(i0)), int2bits(j)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits)
Proof step 18:
int2bits(i) ≠ [] ∧ sgn(bitsand(int2bits(i0), int2bits(j))) ≠ [] ⊦ bitsand(sgn(bitsand(int2bits(i0), int2bits(j))), int2bits(i)) = [], bitsand(sgn(bitsand(int2bits(i), int2bits(i0))), int2bits(j)) = [], sgn(bitsand(sgn(bitsand(int2bits(i0), int2bits(j))), int2bits(i))) = sgn(bitsand(bitsand(int2bits(i), int2bits(i0)), int2bits(j)))

Rule argument

name: and-com rotate: F subst: [bits0, bits] → [sgn(bitsand(int2bits(i0), int2bits(j))), int2bits(i)]

Additional information

Used simplifier rules: and-notempty-09: ⊦ bitsand(int2bits(i), int2bits(i0)) ≠ [];% ⊦ int2bits(j) ≠ [];% ⊦ sgn(bits) = [] ↔ bits = []; Used terms: sgn(bitsand(int2bits(i0), int2bits(j))), int2bits(i)