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Proof step 14 from proof abs-fold-del-p (previous step)

# entries = m + 1, tree[p] = dnode(md, sdt), ((abs|(tree, ROOT_INO, p, ino0) * ino0 => mkdir(m, md, m0, m1, entries -- str) * abs(sdt -- str, entries -- str) * D))(dirs), str ∈ entries, p ∈ tree ⊦ entries = ∅, (( ( abs|(tree[p, dnode(md, sdt -- str)], ROOT_INO, p, ino0) * _∃(λ ad. ino0 => ad * abs(sdt -- str, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ )) * D) )(dirs)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
# entries = m + 1, tree[p] = dnode(md, sdt), ((abs|(tree, ROOT_INO, p, ino0) * ino0 => mkdir(m, md, m0, m1, entries -- str) * abs(sdt -- str, entries -- str) * D))(dirs), p ∈ tree, str ∈ entries ⊦ entries = ∅, ∃ ad. ad.meta = md ∧ ad.size = # ad.entries ∧ ((((ino0 => ad * abs(sdt -- str, ad.entries)) * abs|(tree[p, dnode(ad.meta, sdt -- str)], ROOT_INO, p, ino0)) * D))(dirs)

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜bv⌝ (dirs) ↔ bv; (s)% ⊦ ((D ∧ D0))(dirs) ↔ D(dirs) ∧ D0(dirs); (s)%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap]%sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]%and-com: ⊦ (D ∧ D0) = (D0 ∧ D); from specification heap-sep[dir-heap]%and-ass: ⊦ ((D ∧ D0) ∧ D1) = (D ∧ D0 ∧ D1); from specification heap-sep[dir-heap]% ⊦ (⌜bv⌝ ∧ D) * D0 = (⌜bv⌝ ∧ D * D0); (s)% ⊦ ((_∃ adD * D))(dirs) ↔ (∃ ad. ((adD(ad) * D))(dirs)); (s)