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Proof step 19 from proof and-011x-neg (previous step)

bitsand(false ' + bits3, true ' + bits2.butlast) + bits2.last ' = false ' + bits1, ∀ bits, bits2, bits1. # bits < #(bits3 + true ') → bitsand(false ' + bits, true ' + bits2) = false ' + bits1 → 2 ^ n→i(# bits) = 1 + bits2uint(bitsinvert(bits2)) % 2 ^ n→i(# bits) + bits2uint(bits1) ∨ false ∈ bits ⊦ bits2 = [], 2 ^ (n→i(# bits3) + 1) = 1 + bits2uint(bits1) + bits2uint(bitsinvert(bits2)) % 2 ^ (n→i(# bits3) + 1), false ∈ bits3, ¬ false ∈ bitsand(false ' + bits3, true ' + bits2.butlast) ∧ bits2.last, ¬ bits2.last ∈ bits1 ∧ bits2.last, bits2.last ∧ ¬ false ∈ bitsand(false ' + bits3, true ' + bits2.butlast)

User applied rule insert elim lemma

And obtained the following new goals

Proof step 20:
bits2 = bits0 + boolvar ', bitsand(false ' + bits3, true ' + bits0) + boolvar ' = false ' + bits1, ∀ bits, bits2, bits1. # bits < #(bits3 + true ') → bitsand(false ' + bits, true ' + bits2) = false ' + bits1 → 2 ^ n→i(# bits) = 1 + bits2uint(bitsinvert(bits2)) % 2 ^ n→i(# bits) + bits2uint(bits1) ∨ false ∈ bits ⊦ bits2 = [], 2 ^ (n→i(# bits3) + 1) = 1 + bits2uint(bits1) + bits2uint(bitsinvert(bits2)) % 2 ^ (n→i(# bits3) + 1), false ∈ bits3, ¬ false ∈ bitsand(false ' + bits3, true ' + bits0) ∧ boolvar, ¬ boolvar ∈ bits1 ∧ boolvar, boolvar ∧ ¬ false ∈ bitsand(false ' + bits3, true ' + bits0)

Rule argument

seq: bits ≠ [] ⊦ (boolvar ↔ bits.last) ∧ bits0 = bits.butlast ↔ bits = bits0 + boolvar ' subst: [bits, boolvar, bits0] → [bits2, boolvar, bits0] rotate: T spec: list-last inst: bitlista name: last-elim

Additional information

Used simplifier rules: bits ≠ [] ⊦ (boolvar ↔ bits.last) ∧ bits0 = bits.butlast ↔ bits = bits0 + boolvar '; (elim) Used terms: bits2, boolvar, bits0