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Proof step 7 from proof dups-eqsize-entries (previous step)

∀ key. key.dentry? ∧ key.ino = ino → ¬ key.name ∈ entries, ∀ ks0, ino, entries. # ks0 < # ∅ → (∀ key. key ∈ ks0 → key.dentry? ∧ key.ino = ino) ∧ (∀ key. key.dentry? ∧ key.ino = ino → (key ∈ ks0 ↔ key.name ∈ entries)) → # ks0 = # entries ⊦ entries = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ entries = entries0 ↔ (∀ str. (str ∈ entries ↔ str ∈ entries0) ∧ (str ∈ entries → entries[str] = entries0[str]))
Proof step 9:
∀ key. key.dentry? ∧ key.ino = ino → ¬ key.name ∈ entries, ∀ ks0, ino, entries. # ks0 < # ∅ → (∀ key. key ∈ ks0 → key.dentry? ∧ key.ino = ino) ∧ (∀ key. key.dentry? ∧ key.ino = ino → (key ∈ ks0 ↔ key.name ∈ entries)) → # ks0 = # entries ⊦ ∀ str. (str ∈ entries ↔ str ∈ ∅) ∧ (str ∈ entries → entries[str] = (∅)[str])

Rule argument

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

Additional information

Used terms: ∅, entries