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Proof step 8 from proof count-entries (previous step)

∀ str. str ∈ entries ↔ str ∈ sdt, ∀ entries0, sdt. # entries0 < # entries → (∀ str. str ∈ entries0 ↔ str ∈ sdt) → # entries0 = # sdt ⊦ # entries = # sdt, entries = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ entries = ∅ ↔ ¬(∃ str, ino, entries0. entries = entries0[str, ino] ∧ ¬ str ∈ entries0)
Proof step 10:
∀ str. str ∈ entries ↔ str ∈ sdt, ∀ entries0, sdt. # entries0 < # entries → (∀ str. str ∈ entries0 ↔ str ∈ sdt) → # entries0 = # sdt ⊦ # entries = # sdt, ∀ str, ino, entries0. ¬(entries = entries0[str, ino] ∧ ¬ str ∈ entries0)

Rule argument

spec: store, inst: dir-entries name: exnotempty seq: ⊦ entries = ∅ ↔ ¬ (∃ str, ino, entries0. entries = entries0[str, ino] ∧ ¬ str ∈ entries0) subst: [entries] → [entries] flags: <all paths>

Additional information

Used terms: entries