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Proof step 25 from proof path-split-wand (previous step)

((abs(tree, ino) * abs(sdt0, entries0) * ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino])))(dirs), p0 ∈ tree, ∀ tree, ino0, dirs. abs(tree, ino0)(dirs) ∧ p0 ∈ tree → (∃ ino. ((abs(tree[p0], ino) * (abs(tree[p0], ino) -* abs(tree, ino0))))(dirs)) ⊦ str ∈ sdt0, str ∈ entries0, ∃ ino. (( abs(tree[p0], ino) * ( abs(tree[p0], ino) -* _∃ (λ ad. (⌜str ∈ ad.entries⌝ ∧ (abs(tree, ad.entries[str]) * abs(sdt0, ad.entries -- str)) * ino0 => ad) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ ))) )(dirs)

User applied rule apply lemma

And obtained the following new goals

Proof step 26:
⊦ ∀ D. ((D0 * D))(dirs) → (∃ dirs0, dirs1. dirs0 ⊥ dirs1 ∧ dirs0 ∪ dirs1 = dirs ∧ D0(dirs0) ∧ D(dirs1))
Proof step 27:
((abs(tree, ino) * abs(sdt0, entries0) * ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino])))(dirs), p0 ∈ tree, ∀ tree, ino0, dirs. abs(tree, ino0)(dirs) ∧ p0 ∈ tree → (∃ ino. ((abs(tree[p0], ino) * (abs(tree[p0], ino) -* abs(tree, ino0))))(dirs)), ∀ D. ((abs(tree, ino) * D))(dirs) → (∃ dirs0, dirs1. dirs0 ⊥ dirs1 ∧ dirs0 ∪ dirs1 = dirs ∧ abs(tree, ino)(dirs0) ∧ D(dirs1)) ⊦ str ∈ entries0, str ∈ sdt0, ∃ ino. (( abs(tree[p0], ino) * ( abs(tree[p0], ino) -* _∃ (λ ad. (⌜str ∈ ad.entries⌝ ∧ (abs(tree, ad.entries[str]) * abs(sdt0, ad.entries -- str)) * ino0 => ad) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ ))) )(dirs)

Rule argument

spec: heap-sep, inst: dir-heap name: sep-split-interactive seq: ⊦ ∀ D. ((D0 * D))(dirs) → (∃ dirs0, dirs1. dirs0 ⊥ dirs1 ∧ dirs0 ∪ dirs1 = dirs ∧ D0(dirs0) ∧ D(dirs1)) subst: [D0, dirs] → [abs(tree, ino), dirs]

Additional information

Used terms: abs(tree, ino), dirs