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Proof step 12 from proof ri-cons-drop (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
key-cons(key, fs[ri[key]]), ri[key] ∈ as, key ∈ ri, ri[key] ∈ fs, ∀ adr. ¬ adr ∈ as ∩ range ri ⊦ # as = 0, as = ∅, key-cons(key, (∅)[ri[key]])

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[node-store]%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]% ⊦ adr ∈ as → (fs \ as)[adr] = (∅)[adr]; (s)