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Proof step 9 from proof abs-path-h (previous step)

((ino => mkdir(# entries, md, m0, m1, entries) * abs(sdt, entries) * D))(dirs), ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ tree, ino, D, dirs, files. ((abs(tree, ino) * D))(dirs) ∧ (∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (ispath(p0, ino, dirs, files) ↔ p0 ∈ tree), ∀ ino0. ⌜ino0⌝ ∈ sdt.fids → ino0 ∈ files ⊦ str ∈ entries ∧ ispath(p0, entries[str], dirs, files) ↔ str ∈ sdt ∧ p0 ∈ sdt[str]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ str ∈ sdt → abs(sdt, sdi) = (⌜str ∈ sdi⌝ ∧ abs(sdt -- str, sdi -- str) * abs(sdt[str], sdi[str]))
Proof step 11:
⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt
Proof step 12:
((ino => mkdir(# entries, md, m0, m1, entries) * (⌜str ∈ entries⌝ ∧ abs(sdt -- str, entries -- str) * abs(sdt[str], entries[str])) * D))(dirs), str ∈ sdt, ∀ ino0. ⌜ino0⌝ ∈ sdt.fids → ino0 ∈ files, ∀ tree, ino, D, dirs, files. ((abs(tree, ino) * D))(dirs) ∧ (∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (ispath(p0, ino, dirs, files) ↔ p0 ∈ tree), ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs ⊦ str ∈ entries ∧ ispath(p0, entries[str], dirs, files) ↔ str ∈ sdt ∧ p0 ∈ sdt[str]

Rule argument

spec: <toplevel> 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: entries-dom1: ⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt;%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] Used terms: str, entries, sdt