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Proof step 30 from proof and-size-one (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ bits ≠ [] → ((bits + bits0).first ↔ bits.first)

Rule argument

seq: ⊦ bits ≠ [] → ((bits + bits0).first ↔ bits.first) subst: [bits0, bits] → [boolvar0 ', bits4] rotate: F spec: list inst: bitlista name: first-01

Additional information

Used simplifier rules: ⊦ [] + bits = bits;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ boolvar ' = boolvar0 ' ↔ boolvar ↔ boolvar0;% ⊦ boolvar '.first ↔ boolvar;% ⊦ # [] = 0;% ⊦ 0 ≤ m; Used terms: boolvar0 ', bits4