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Proof step 17 from proof band-min-01 (previous step)

# int2bits(i) < # bitsand(int2bits(i), int2bits(j)), # int2bits(j) < # bitsand(int2bits(i), int2bits(j)), j < 0, i < 0, i0 + 2 ^ n→i(# int2bits(j)) ≤ 0, i0 + 2 ^ n→i(# int2bits(i)) ≤ 0, bits2int(bitsand(int2bits(i), int2bits(j))) < i0, # int2bits(i) < # int2bits(j), 2 ^ n→i(# int2bits(j)) + bits2int(bitsand(int2bits(i), int2bits(j))) < 0, # int2bits(i) + 1 < # bitsand(int2bits(i), int2bits(j)) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits)
Proof step 19:
int2bits(i) ≠ [] ∧ int2bits(j) ≠ [], # int2bits(i) < # bitsand(int2bits(j), int2bits(i)), # int2bits(j) < # bitsand(int2bits(j), int2bits(i)), j < 0, i < 0, i0 + 2 ^ n→i(# int2bits(j)) ≤ 0, i0 + 2 ^ n→i(# int2bits(i)) ≤ 0, bits2int(bitsand(int2bits(j), int2bits(i))) < i0, # int2bits(i) < # int2bits(j), 2 ^ n→i(# int2bits(j)) + bits2int(bitsand(int2bits(j), int2bits(i))) < 0, # int2bits(i) + 1 < # bitsand(int2bits(j), int2bits(i)) ⊦

Rule argument

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

Additional information

Used simplifier rules: ⊦ int2bits(j) ≠ []; Used terms: int2bits(j), int2bits(i)