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Proof step 9 from proof index-update-dom (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ as ⊆ as0 ↔ (∀ adr. adr ∈ as → adr ∈ as0)
Proof step 11:
# 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

Rule argument

spec: set-basic, inst: node-store name: Subset seq: ⊦ as ⊆ as0 ↔ (∀ adr. adr ∈ as → adr ∈ as0) subst: [as0, as] → [range ri, ax0.toSet] flags: <all paths>

Additional information

Used terms: range ri, ax0.toSet