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Proof step 9 from proof ri-cons-drop (previous step)

key-cons(key, fs[ri[key]]), key ∈ ri, ri[key] ∈ fs, ∀ adr. adr ∈ as ∩ range ri ↔ adr ∈ ∅ ⊦ key-cons(key, (fs \ as)[ri[key]])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
key-cons(key, fs[ri[key]]), ri[key] ∈ fs, key ∈ ri, ∀ adr. ¬ adr ∈ as ∩ range ri ⊦ key-cons(key, (fs \ as)[ri[key]])

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ adr ∈ ∅; (s)