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Proof step 38 from proof path-split-wand (previous step)

((abs(tree, ino) * abs(sdt0, entries0) * ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino])))(dirs2 ∪ dirs ∪ dirs1), ((ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino]) * abs(sdt0, entries0)))(dirs1), ((abs(tree[p0], ino1) -* abs(tree, ino)))(dirs), abs(tree, ino)(dirs2 ∪ dirs), abs(tree[p0], ino1)(dirs2), p0 ∈ tree, dirs1 ⊥ dirs2, dirs2 ⊥ dirs, dirs1 ⊥ dirs ⊦ (( abs(tree[p0], ino1) * ( abs(tree[p0], ino1) -* _∃ (λ ad. (⌜str ∈ ad.entries⌝ ∧ (abs(tree, ad.entries[str]) * abs(sdt0, ad.entries -- str)) * ino0 => ad) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ ))) )(dirs2 ∪ dirs ∪ dirs1), str ∈ entries0, str ∈ sdt0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ D(dirs) ∧ D0(dirs0) ∧ dirs ⊥ dirs0 → ((D * D0))(dirs ∪ dirs0)
Proof step 40:
((ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino]) * abs(sdt0, entries0)))(dirs1), ((abs(tree, ino) * abs(sdt0, entries0) * ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino])))(dirs2 ∪ dirs ∪ dirs1), ((abs(tree[p0], ino1) -* abs(tree, ino)))(dirs), abs(tree[p0], ino1)(dirs2), abs(tree, ino)(dirs2 ∪ dirs), p0 ∈ tree, dirs1 ⊥ dirs, dirs2 ⊥ dirs, dirs1 ⊥ dirs2 ⊦ (( abs(tree[p0], ino1) * ( abs(tree[p0], ino1) -* _∃ (λ ad. (⌜str ∈ ad.entries⌝ ∧ (abs(tree, ad.entries[str]) * abs(sdt0, ad.entries -- str)) * ino0 => ad) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ ))) )(dirs2 ∪ dirs ∪ dirs1), str ∈ sdt0, str ∈ entries0, abs(tree[p0], ino1)(dirs2) ∧ (( abs(tree[p0], ino1) -* _∃ (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 ∪ dirs1) ∧ dirs2 ⊥ dirs ∪ dirs1

Rule argument

spec: heap-sep, inst: dir-heap name: split-union seq: ⊦ D(dirs) ∧ D0(dirs0) ∧ dirs ⊥ dirs0 → ((D * D0))(dirs ∪ dirs0) subst: [dirs0, dirs, D0, D] → [dirs ∪ dirs1, dirs2, abs(tree[p0], ino1) -* _∃ (λ ad. ( ⌜str ∈ ad.entries⌝ ∧ ( abs(tree, ad.entries[str]) * abs(sdt0, ad.entries -- str)) * ino0 => ad) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ ), abs(tree[p0], ino1)] flags: <all paths>

Additional information

Used terms: dirs ∪ dirs1, dirs2, abs(tree[p0], ino1) -* _∃ (λ ad. ( ⌜str ∈ ad.entries⌝ ∧ ( abs(tree, ad.entries[str]) * abs(sdt0, ad.entries -- str)) * ino0 => ad) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ ), abs(tree[p0], ino1)