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Proof step 12 from proof abs-delorphans (previous step)

links(ino, dirs) = ∅, abs(tree, ROOT_INO)(dirs), ispath(p, ROOT_INO, ino, dirs, files), ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs ⊦ # orphans(dirs, files) = 0, ino = ROOT_INO, orphans(dirs, files) = ∅

User applied rule apply lemma

And obtained the following new goals

Proof step 13:
∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ((abs(tree, ino) * D))(dirs) ⊦ ispath(p, ino, dirs, files) ↔ p ∈ tree
Proof step 14:
ispath(p, ROOT_INO, dirs, files) ↔ p ∈ tree, links(ino, dirs) = ∅, abs(tree, ROOT_INO)(dirs), ispath(p, ROOT_INO, ino, dirs, files), ino ∈ dirs, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files ⊦ # orphans(dirs, files) = 0, ino = ROOT_INO, orphans(dirs, files) = ∅

Rule argument

spec: <toplevel> name: abs-path-h seq: ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ((abs(tree, ino) * D))(dirs) ⊦ ispath(p, ino, dirs, files) ↔ p ∈ tree subst: [files, dirs, tree, ino, D, p] → [files, dirs, tree, ROOT_INO, emp, p]

Additional information

Used simplifier rules: ⊦ emp * D = D; (s)%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: files, dirs, tree, ROOT_INO, emp, p