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Proof step 15 from proof abs-has-rootpath-h (previous step)

(((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt, entries)) * {dirs0} ))(dirs), ino ∈ dirs, entries ⊆ entries0, ∀ str. str ∈ sdt → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files))), ∀ sdt0, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files. # sdt0 < # sdt → (((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt0, entries)) * {dirs0} ))(dirs) ∧ ino ∈ dirs ∧ entries ⊆ entries0 ∧ (∀ str. str ∈ sdt0 → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt0[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)))) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)) ⊦ ino ∈ dirs0, ∃ p. ispath(p, ino0, ino, dirs, files)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 16:
sdt ≠ ∅ ⊦ ((abs(sdt, sdi) * D))(dirs) ↔ (∃ tree, ino, str, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs))
Proof step 17:
sdt = ∅, (((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt, entries)) * {dirs0} ))(dirs), ino ∈ dirs, entries ⊆ entries0, ∀ str. str ∈ sdt → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files))), ∀ sdt0, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files. # sdt0 < # sdt → (((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt0, entries)) * {dirs0} ))(dirs) ∧ ino ∈ dirs ∧ entries ⊆ entries0 ∧ (∀ str. str ∈ sdt0 → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt0[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)))) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)) ⊦ ino ∈ dirs0, ∃ p. ispath(p, ino0, ino, dirs, files)
Proof step 26:
ino ∈ dirs, entries ⊆ entries0, ∀ str. str ∈ sdt → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files))), ∃ tree, ino, str, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ entries = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * {dirs0} * ino0 => mkdir(n0, md, m0, n, entries0)))(dirs), ∀ sdt0, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files. # sdt0 < # sdt → (((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt0, entries)) * {dirs0} ))(dirs) ∧ ino ∈ dirs ∧ entries ⊆ entries0 ∧ (∀ str. str ∈ sdt0 → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt0[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)))) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)) ⊦ sdt = ∅, ino ∈ dirs0, ∃ p. ispath(p, ino0, ino, dirs, files)

Rule argument

spec: <toplevel> name: entries-any seq: sdt ≠ ∅ ⊦ ((abs(sdt, sdi) * D))(dirs) ↔ (∃ tree, ino, str, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs)) subst: [dirs, D, sdi, sdt] → [dirs, {dirs0} * ino0 => mkdir(n0, md, m0, n, entries0), entries, sdt] flags: <all paths>

Additional information

Used simplifier rules: 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: dirs, {dirs0} * ino0 => mkdir(n0, md, m0, n, entries0), entries, sdt