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Proof step 19 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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
bits ≠ [], bits.first ↔ boolvar ⊦ sgn(boolvar ' + bits) = sgn(bits)
Proof step 21:
¬(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
Proof step 34:
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(bitsinvert(bits1) + true ' + bits0).rest

Rule argument

seq: bits ≠ [], bits.first ↔ boolvar ⊦ sgn(boolvar ' + bits) = sgn(bits) subst: [bits, boolvar] → [bitsinvert(bits1) + true ' + bits0, true] rotate: F spec: bitopsaux inst: name: sgn-rec-01

Additional information

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