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Proof step 527 from proof bitsand-cast (previous step)

# bits1 = 2 + # bits, sgn(mklist(true, # bits0) + true ' + false ' + bits) = true ' + false ' + bits, 2 + # bits + # bits0 ≤ # bitsand(bits2, bits4) + # bitsand(mklist(true, # bits0), bits0) + # bits1 ⊦ bits0 = [], bits4 + bits0 = [], bits2 + mklist(true, # bits0) = [], lastn(2 + # bits + # bits0, fillsgn(2 + # bits + # bits0, (bits4 + bits0) + bitsand(bits1, true ' + false ' + bits))) = lastn(2 + # bits + # bits0, (bitsand(bits2, bits4) + bitsand(mklist(true, # bits0), bits0)) + bitsand(bits1, true ' + false ' + bits)), bits1 = [], (bits4 + bits0) + bits1 = [], bits2 = [], bits4 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 528:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits)
Proof step 529:
mklist(true, # bits0) ≠ [] ∧ bits0 ≠ [], # bits1 = 2 + # bits, sgn(mklist(true, # bits0) + true ' + false ' + bits) = true ' + false ' + bits, 2 + # bits + # bits0 ≤ # bitsand(bits2, bits4) + # bitsand(bits0, mklist(true, # bits0)) + # bits1 ⊦ bits0 = [], bits4 + bits0 = [], bits2 + mklist(true, # bits0) = [], lastn(2 + # bits + # bits0, fillsgn(2 + # bits + # bits0, (bits4 + bits0) + bitsand(bits1, true ' + false ' + bits))) = lastn(2 + # bits + # bits0, (bitsand(bits2, bits4) + bitsand(bits0, mklist(true, # bits0))) + bitsand(bits1, true ' + false ' + bits)), bits1 = [], (bits4 + bits0) + bits1 = [], bits2 = [], bits4 = []

Rule argument

name: and-com rotate: F subst: [bits0, bits] → [bits0, mklist(true, # bits0)]

Additional information

Used simplifier rules: ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ mklist(boolvar, m) = [] ↔ m = 0;% ⊦ # bits = 0 ↔ bits = []; Used terms: bits0, mklist(true, # bits0)