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Proof step 11 from proof index-update-dom (previous step)

# ax0 = # ax1, adr ∈ ax0, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∀ adr. adr ∈ ax0.toSet → adr ∈ range ri ⊦ fs[adr].key ∈ ri

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
# ax0 = # ax1, adr ∈ ax0, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∀ adr0. adr0 ∈ ax0 → adr0 ∈ ri ⊦ fs[adr].key ∈ ri

Rule argument

Additional information

Used simplifier rules: ⊦ adr ∈ range ri ↔ adr ∈ ri; (s)% ⊦ adr ∈ ax.toSet ↔ adr ∈ ax; (s)