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Proof step 5 from proof abs-has-rootpath-h (previous step)

⊦ (∀ fid. ∀ ino1, dirs, tree, ino0, dirs0, files. fnode(fid) = tree ∧ ino1 ∈ dirs ∧ ((abs(tree, ino0) * {dirs0} ))(dirs) → ino1 ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino1, dirs, files))) ∧ (∀ md, sdt. (∀ str. str ∈ sdt → (∀ ino1, dirs, tree, ino0, dirs0, files. sdt[str] = tree ∧ ino1 ∈ dirs ∧ ((abs(tree, ino0) * {dirs0} ))(dirs) → ino1 ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino1, dirs, files)))) → (∀ ino1, dirs, tree, ino0, dirs0, files. dnode(md, sdt) = tree ∧ ino1 ∈ dirs ∧ ((abs(tree, ino0) * {dirs0} ))(dirs) → ino1 ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino1, dirs, files))))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
⊦ abs(fnode(fid), ino) = (emp ∧ ⌜ino = fid.ino⌝ )
Proof step 7:
⊦ abs(dnode(md, sdt), ino) = _∃(λ ad. ino => ad * abs(sdt, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ )
Proof step 8:
ad.size = # ad.entries, (((ino0 => ad * abs(sdt, ad.entries)) * {dirs0} ))(dirs), ino ∈ dirs, ∀ str. str ∈ sdt → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files))) ⊦ ino ∈ dirs0, ∃ p. ispath(p, ino0, ino, dirs, files)

Rule argument

Additional information

Used simplifier rules: abs-fnode: ⊦ abs(fnode(fid), ino) = (emp ∧ ⌜ino = fid.ino⌝ );%abs-dnode: ⊦ abs(dnode(md, sdt), ino) = _∃ (λ ad. ino => ad * abs(sdt, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ );% ⊦ ((_∃ adD * D))(dirs) ↔ (∃ ad. ((adD(ad) * D))(dirs)); (s)% ⊦ {dirs} (dirs0) ↔ dirs = dirs0; (s)% ⊦ ⌜bv⌝ (dirs) ↔ bv; (s)% ⊦ ((D ∧ D0))(dirs) ↔ D(dirs) ∧ D0(dirs); (s)%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]%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]% ⊦ emp * D = D; (s)% ⊦ (⌜bv⌝ ∧ D) * D0 = (⌜bv⌝ ∧ D * D0); (s)