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Proof step 829 from proof bitsand-cast (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 830:
⊦ bits ≠ [] ∧ bits0 ≠ [] → bitsand(bits, bits0) = bitsand(bits0, bits)
Proof step 831:
false ' + true ' + bits ≠ [] ∧ bits1 ≠ [], 2 + # bits + # bits0 = # bitsand(mklist(false, # bits0), bits0) + # bitsand(bits1, false ' + true ' + bits), sgn(mklist(false, # bits0) + false ' + true ' + bits) = false ' + true ' + bits, 2 + # bits = # bits1, # bits1 = # bitsand(bits1, false ' + true ' + bits) ⊦ bits0 = [], (bits4 + bits0) + bits1 = [], bits4 + bits0 = [], bits2 + mklist(false, # bits0) = [], bits1 = [], mklist(false, # bits0) + bitsand(bits1, false ' + true ' + bits) = bitsand(mklist(false, # bits0), bits0) + bitsand(bits1, false ' + true ' + bits), bits2 = [], bits4 = []

Rule argument

name: and-com rotate: F subst: [bits0, bits] → [bits1, false ' + true ' + bits]

Additional information

Used simplifier rules: ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ boolvar ' ≠ [];% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1; Used terms: bits1, false ' + true ' + bits