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Proof step 34 from proof and-ass (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 35:
⊦ bits0 ≠ [] ∧ bits = true ' → bitsand(bits, bits0) = bits0
Proof step 36:
bitsand(bits3, bits4) + (boolvar ∧ boolvar0)' ≠ [] ∧ bits.first ' = true ', ∀ bits2, bits0, bits1. # bits2 < # bits.first ' → true → bits2 = [] ∨ bits0 = [] ∨ bits1 = [] ∨ bitsand(bits2, bitsand(bits0, bits1)) = bitsand(bitsand(bits2, bits0), bits1) ⊦ bitsand(bits3, bits4) + (boolvar ∧ boolvar0)' = bitsand(bitsand(bits.first ', bits3 + boolvar '), bits4 + boolvar0 '), bits3 + boolvar ' = (bits3 + boolvar ').first ', bits4 + boolvar0 ' = (bits4 + boolvar0 ').first '

Rule argument

name: and-rec-03 rotate: F subst: [bits0, bits] → [bitsand(bits3, bits4) + (boolvar ∧ boolvar0)', bits.first ']

Additional information

Used simplifier rules: bitsand-zerol: ⊦ bitsand(false ', bits + boolvar ') = false ';% ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ boolvar ' ≠ [];% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ boolvar ' = boolvar0 ' ↔ boolvar ↔ boolvar0; Used terms: bitsand(bits3, bits4) + (boolvar ∧ boolvar0)', bits.first '