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Proof step 25 from proof int2bits-falsein-01 (previous step)

uint2bits(~ i) = (false ' + bits1) + true ' + bits0, bitsinvert(uint2bits(~ i)) +1 = (true ' + bitsinvert(bits1)) + true ' + bits0, i < 0, i < ~1, false ∈ uint2bits(~ i) ⊦ bits1 = [], true ∈ bits0, false ∈ sgn((true ' + bitsinvert(bits1)) + true ' + bits0).rest, bitsinvert(bits1).first

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 26:
⊦ bits ≠ [] ∧ (bits.first ↔ ¬ boolvar) → sgn(boolvar ' + bits) = boolvar ' + bits
Proof step 27:
¬(bitsinvert(bits1) + true ' + bits0 ≠ [] ∧ ((bitsinvert(bits1) + true ' + bits0).first ↔ ¬ true)), uint2bits(~ i) = (false ' + bits1) + true ' + bits0, bitsinvert(uint2bits(~ i)) +1 = (true ' + bitsinvert(bits1)) + true ' + bits0, i < 0, i < ~1, false ∈ uint2bits(~ i) ⊦ bits1 = [], true ∈ bits0, false ∈ sgn((true ' + bitsinvert(bits1)) + true ' + bits0).rest, bitsinvert(bits1).first
Proof step 30:
bitsinvert(bits1) + true ' + bits0 ≠ [] ∧ ((bitsinvert(bits1) + true ' + bits0).first ↔ ¬ true), uint2bits(~ i) = (false ' + bits1) + true ' + bits0, bitsinvert(uint2bits(~ i)) +1 = (true ' + bitsinvert(bits1)) + true ' + bits0, i < 0, i < ~1, false ∈ uint2bits(~ i) ⊦ bits1 = [], true ∈ bits0, false ∈ (true ' + bitsinvert(bits1) + true ' + bits0).rest, bitsinvert(bits1).first

Rule argument

seq: ⊦ bits ≠ [] ∧ (bits.first ↔ ¬ boolvar) → sgn(boolvar ' + bits) = boolvar ' + bits subst: [bits, boolvar] → [bitsinvert(bits1) + true ' + bits0, true] rotate: F spec: bitopsaux inst: name: sgn-base-03

Additional information

Used simplifier rules: ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1; Used terms: bitsinvert(bits1) + true ' + bits0, true