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Proof step 21 from proof abs-has-rootpath-h (previous step)

((ino0 => mkdir(n0, md, m0, n, entries0) * {dirs0} ))(dirs), ino ∈ dirs, ∀ sdt0, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files. # sdt0 < # ∅ → (((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 = ino0, ino ∈ dirs0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ ((D * D0))(dirs) ↔ (∃ dirs0, dirs1. dirs0 ⊥ dirs1 ∧ dirs = dirs0 ∪ dirs1 ∧ D(dirs0) ∧ D0(dirs1))
Proof step 23:
ino ∈ dirs, ∃ dirs2, dirs1. dirs2 ⊥ dirs1 ∧ dirs = dirs2 ∪ dirs1 ∧ ((ino0 => mkdir(n0, md, m0, n, entries0)))(dirs2) ∧ {dirs0} (dirs1), ∀ sdt0, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files. # sdt0 < # ∅ → (((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 = ino0, ino ∈ dirs0

Rule argument

spec: heap-sep, inst: dir-heap name: sep seq: ⊦ ((D * D0))(dirs) ↔ (∃ dirs0, dirs1. dirs0 ⊥ dirs1 ∧ dirs = dirs0 ∪ dirs1 ∧ D(dirs0) ∧ D0(dirs1)) subst: [dirs, D0, D] → [dirs, {dirs0} , ino0 => mkdir(n0, md, m0, n, entries0)] flags: <all paths>

Additional information

Used terms: dirs, {dirs0} , ino0 => mkdir(n0, md, m0, n, entries0)