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Proof step 25 from proof lastn-fill-sgn (previous step)

# bits0 = # fillsgn(# bits0, sgn(bits1 + bits0)), # sgn(bits1 + bits0) < # bits1 + # bits0, # sgn(bits1 + bits0) < # bits0, # sgn(bits1 + bits0) + 1 < # bits1 + # bits0 ⊦ bits0 = [], fillsgn(# bits0, sgn(bits1 + bits0)) = bits0, bits1 + bits0 = [], bits1 = [], true ∈ bits1 ∧ false ∈ bits1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 26:
⊦ ¬ true ∈ bits ∧ bits ≠ [] → sgn(bits + bits0) = sgn(false ' + bits0)
Proof step 27:
¬(¬ true ∈ bits1 ∧ bits1 ≠ []), # bits0 = # fillsgn(# bits0, sgn(bits1 + bits0)), # sgn(bits1 + bits0) < # bits1 + # bits0, # sgn(bits1 + bits0) < # bits0, # sgn(bits1 + bits0) + 1 < # bits1 + # bits0 ⊦ bits0 = [], fillsgn(# bits0, sgn(bits1 + bits0)) = bits0, bits1 + bits0 = [], bits1 = [], true ∈ bits1 ∧ false ∈ bits1
Proof step 37:
¬ true ∈ bits1 ∧ bits1 ≠ [], # bits0 = # fillsgn(# bits0, sgn(false ' + bits0)), # sgn(false ' + bits0) < # bits1 + # bits0, # sgn(false ' + bits0) < # bits0, # sgn(false ' + bits0) + 1 < # bits1 + # bits0 ⊦ bits0 = [], fillsgn(# bits0, sgn(false ' + bits0)) = bits0, bits1 + bits0 = [], bits1 = [], true ∈ bits1 ∧ false ∈ bits1

Rule argument

name: sgn-append-04 rotate: F subst: [bits0, bits] → [bits0, bits1]

Additional information

Used terms: bits0, bits1