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Proof step 820 from proof bitsand-cast (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 821:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits)
Proof step 822:
mklist(false, # bits0) ≠ [] ∧ bits0 ≠ [], # bits1 = # bitsand(bits1, false ' + true ' + bits), 2 + # bits = # bits1, sgn(mklist(false, # bits0) + false ' + true ' + bits) = false ' + true ' + bits, 2 + # bits + # bits0 ≤ # bitsand(bits2, bits4) + # bitsand(bits0, mklist(false, # bits0)) + # bits1, 2 + # bits + # bits0 < # bitsand(bits0, mklist(false, # bits0)) + # bits1 ⊦ bits1 = [], bits4 = [], bits2 = [], mklist(false, # bits0) + bitsand(bits1, false ' + true ' + bits) = lastn(2 + # bits + # bits0, bitsand(bits0, mklist(false, # bits0)) + bitsand(bits1, false ' + true ' + bits)), bits2 + mklist(false, # bits0) = [], bits4 + bits0 = [], (bits4 + bits0) + bits1 = [], bits0 = [], 2 + # bits + # bits0 = # bitsand(bits0, mklist(false, # bits0)) + # bits1

Rule argument

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