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Proof step 3 from proof abs-unfold-entry-p (previous step)

entries[str] ∈ dirs0, str ∈ entries, dirs1 ≡ dirs0[entries[str], ad] ⊦ ((abs(sdt, entries) * D))(dirs0) ↔ str ∈ sdt ∧ ((abs(sdt[str], entries[str]) * abs(sdt -- str, entries -- str) * D))(dirs0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ str ∈ sdt → abs(sdt, sdi) = (⌜str ∈ sdi⌝ ∧ abs(sdt -- str, sdi -- str) * abs(sdt[str], sdi[str]))

Rule argument

spec: vfs-abs name: entries-recdel seq: ⊦ str ∈ sdt → abs(sdt, sdi) = ( ⌜str ∈ sdi⌝ ∧ abs(sdt -- str, sdi -- str) * abs(sdt[str], sdi[str])) subst: [str, sdi, sdt] → [str, entries, sdt] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ⌜true⌝ = true; (s)% ⊦ (true ∧ D) = D; (s)%and-ass: ⊦ ((D ∧ D0) ∧ D1) = (D ∧ D0 ∧ D1); from specification heap-sep[dir-heap]%and-com: ⊦ (D ∧ D0) = (D0 ∧ D); from specification heap-sep[dir-heap]%sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap]%entries-dom1: ⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt; (forward) from specification vfs-abs Used terms: str, entries, sdt