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Proof step 4 from proof abs-is-file (previous step)

((abs(tree, ino) * D))(dirs) ⊦ ino ∈ dirs, tree = fnode(⌜ino⌝ ) ∧ D(dirs)

User applied rule constructor cut

And obtained the following new goals

Proof step 5:
⊦ abs(dnode(md, sdt), ino) = _∃(λ ad. ino => ad * abs(sdt, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ )
Proof step 6:
tree = fnode(fid), ((abs(tree, ino) * D))(dirs) ⊦ ino ∈ dirs, tree = fnode(⌜ino⌝ ) ∧ D(dirs)

Rule argument

var: tree

Additional information

Used simplifier rules: abs-dnode: ⊦ abs(dnode(md, sdt), ino) = _∃ (λ ad. ino => ad * abs(sdt, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ );%directory-tree, entries-tree generated by ∅, dnode, fnode, [ ];;% ⊦ ((_∃ adD * D))(dirs) ↔ (∃ ad. ((adD(ad) * D))(dirs)); (s)% ⊦ (⌜bv⌝ ∧ D) * D0 = (⌜bv⌝ ∧ D * D0); (s)%and-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]%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]% ⊦ ((D ∧ D0))(dirs) ↔ D(dirs) ∧ D0(dirs); (s)% ⊦ ⌜bv⌝ (dirs) ↔ bv; (s)% ⊦ ((ino => ad * D))(dirs) → ino ∈ dirs; (s)