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Proof step 52 from proof and-ass (previous step)

∀ bits2, bits0, bits1. # bits2 < #(bits4 + boolvar0 ') → true → bits2 = [] ∨ bits0 = [] ∨ bits1 = [] ∨ bitsand(bits2, bitsand(bits0, bits1)) = bitsand(bitsand(bits2, bits0), bits1) ⊦ bits + boolvar1 ' = (bits + boolvar1 ').first ', bits3 + boolvar ' = (bits3 + boolvar ').first ', bitsand(bits4, bitsand(bits, bits3)) + (boolvar0 ∧ boolvar1 ∧ boolvar)' = bitsand(bitsand(bits4, bits) + (boolvar0 ∧ boolvar1)', bits3 + boolvar '), bits4 = [], bits4 + boolvar0 ' = bits4.first ', bitsand(bits, bits3) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 53:
⊦ 1 < # bits ∧ 1 < # bits0 → bitsand(bits, bits0) = bitsand(bits.butlast, bits0.butlast) + (bits.last ∧ bits0.last)
Proof step 54:
¬(1 < #(bitsand(bits4, bits) + (boolvar0 ∧ boolvar1)') ∧ 1 < #(bits3 + boolvar ')), ∀ bits2, bits0, bits1. # bits2 < #(bits4 + boolvar0 ') → true → bits2 = [] ∨ bits0 = [] ∨ bits1 = [] ∨ bitsand(bits2, bitsand(bits0, bits1)) = bitsand(bitsand(bits2, bits0), bits1) ⊦ bits + boolvar1 ' = (bits + boolvar1 ').first ', bits3 + boolvar ' = (bits3 + boolvar ').first ', bitsand(bits4, bitsand(bits, bits3)) + (boolvar0 ∧ boolvar1 ∧ boolvar)' = bitsand(bitsand(bits4, bits) + (boolvar0 ∧ boolvar1)', bits3 + boolvar '), bits4 = [], bits4 + boolvar0 ' = bits4.first ', bitsand(bits, bits3) = []
Proof step 57:
1 < #(bitsand(bits4, bits) + (boolvar0 ∧ boolvar1)') ∧ 1 < #(bits3 + boolvar '), ∀ bits2, bits0, bits1. # bits2 < #(bits4 + boolvar0 ') → true → bits2 = [] ∨ bits0 = [] ∨ bits1 = [] ∨ bitsand(bits2, bitsand(bits0, bits1)) = bitsand(bitsand(bits2, bits0), bits1) ⊦ bits + boolvar1 ' = (bits + boolvar1 ').first ', bits3 + boolvar ' = (bits3 + boolvar ').first ', bitsand(bits4, bitsand(bits, bits3)) + (boolvar0 ∧ boolvar1 ∧ boolvar)' = bitsand((bitsand(bits4, bits) + (boolvar0 ∧ boolvar1)').butlast, (bits3 + boolvar ').butlast) + ((bitsand(bits4, bits) + (boolvar0 ∧ boolvar1)').last ∧ (bits3 + boolvar ').last), bits4 = [], bits4 + boolvar0 ' = bits4.first ', bitsand(bits, bits3) = []

Rule argument

name: and-rec rotate: F subst: [bits0, bits] → [bits3 + boolvar ', bitsand(bits4, bits) + (boolvar0 ∧ boolvar1)']

Additional information

Used terms: bits3 + boolvar ', bitsand(bits4, bits) + (boolvar0 ∧ boolvar1)'