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Proof step 154 from proof byte-law-and-one (previous step)

int2bits(j) = bits0 + bits, # bits = 8, 8 = # fillsgn(8, int2bits(i)), # int2bits(i) < # bitsand(bits, fillsgn(8, int2bits(i))), i < 0, # int2bits(i) < 8, 8 ≤ # bits0 + # bitsand(bits, fillsgn(8, int2bits(i))), # int2bits(i) < # bits0 + # bitsand(bits, fillsgn(8, int2bits(i))), 8 ≤ # bitsand(bits, fillsgn(8, int2bits(i))) ⊦ fillsgn(8, int2bits(i)) = [], bits0 + bits = [], lastn(8, fillsgn(8, bitsand(int2bits(i), bits0 + bits))) = lastn(8, bitsand(bits, fillsgn(8, int2bits(i)))), bitsand(bits, fillsgn(8, int2bits(i))) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 155:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits)
Proof step 156:
int2bits(i) ≠ [] ∧ bits0 + bits ≠ [], int2bits(j) = bits0 + bits, # bits = 8, 8 = # fillsgn(8, int2bits(i)), # int2bits(i) < # bitsand(bits, fillsgn(8, int2bits(i))), i < 0, # int2bits(i) < 8, 8 ≤ # bits0 + # bitsand(bits, fillsgn(8, int2bits(i))), # int2bits(i) < # bits0 + # bitsand(bits, fillsgn(8, int2bits(i))), 8 ≤ # bitsand(bits, fillsgn(8, int2bits(i))) ⊦ fillsgn(8, int2bits(i)) = [], bits0 + bits = [], lastn(8, fillsgn(8, bitsand(bits0 + bits, int2bits(i)))) = lastn(8, bitsand(bits, fillsgn(8, int2bits(i)))), bitsand(bits, fillsgn(8, int2bits(i))) = []

Rule argument

seq: ⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits) subst: [bits0, bits] → [bits0 + bits, int2bits(i)] rotate: F spec: intops inst: name: and-com

Additional information

Used simplifier rules: ⊦ int2bits(j) ≠ []; Used terms: bits0 + bits, int2bits(i)