[Imprint][Data Privacy Statement][E-Mail]

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Proof step 45 from proof abs-tree (previous step)

((ino => mkdir(# entries, md, m0, m1, entries) * abs(sdt0, entries) * D1))(dirs), (((ino => mkdir(# entries, md, m0, m1, entries) * (⌜str ∈ entries⌝ ∧ abs(sdt -- str, entries -- str) * abs(sdt[str], entries[str]))) * D0) )(dirs), str ∈ sdt, ∀ ino0. ⌜ino0⌝ ∈ sdt.fids → ino0 ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ tree1, ino, D1, dirs, tree0, D0, files. ((abs(tree1, ino) * D1))(dirs) ∧ ((abs(tree0, ino) * D0))(dirs) ∧ (∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files) ∧ (∀ ino. ⌜ino⌝ ∈ tree0.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (p0 ∈ tree0 ↔ p0 ∈ tree1) ∧ (p0 ∈ tree0 → (tree0[p0].dir? ↔ tree1[p0].dir?) ∧ (tree0[p0].dir? ⊃ tree0[p0].meta = tree1[p0].meta;tree0[p0].fid = tree1[p0].fid)), ∀ ino0. ⌜ino0⌝ ∈ sdt0.fids → ino0 ∈ files ⊦ (str ∈ sdt ∧ p0 ∈ sdt[str] ↔ str ∈ sdt0 ∧ p0 ∈ sdt0[str]) ∧ ( str ∈ sdt ∧ p0 ∈ sdt[str] → (sdt[str][p0].dir? ↔ dnode(md, sdt0)[str ' + p0].dir?) ∧ (sdt[str][p0].dir? ⊃ sdt[str][p0].meta = dnode(md, sdt0)[str ' + p0].meta;sdt[str][p0].fid = dnode(md, sdt0)[str ' + p0].fid))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 46:
⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt
Proof step 47:
⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi
Proof step 48:
((((abs(sdt -- str, entries -- str) * abs(sdt[str], entries[str])) * ino => mkdir(# entries, md, m0, m1, entries)) * D0))(dirs), ((ino => mkdir(# entries, md, m0, m1, entries) * abs(sdt0, entries) * D1))(dirs), str ∈ sdt0, str ∈ sdt, str ∈ entries, ∀ ino0. ⌜ino0⌝ ∈ sdt.fids → ino0 ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ tree1, ino, D1, dirs, tree0, D0, files. ((abs(tree1, ino) * D1))(dirs) ∧ ((abs(tree0, ino) * D0))(dirs) ∧ (∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files) ∧ (∀ ino. ⌜ino⌝ ∈ tree0.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (p0 ∈ tree0 ↔ p0 ∈ tree1) ∧ (p0 ∈ tree0 → (tree0[p0].dir? ↔ tree1[p0].dir?) ∧ (tree0[p0].dir? ⊃ tree0[p0].meta = tree1[p0].meta;tree0[p0].fid = tree1[p0].fid)), ∀ ino0. ⌜ino0⌝ ∈ sdt0.fids → ino0 ∈ files ⊦ (p0 ∈ sdt[str] ↔ p0 ∈ sdt0[str]) ∧ ( p0 ∈ sdt[str] → (sdt[str][p0].dir? ↔ sdt0[str][p0].dir?) ∧ (sdt[str][p0].dir? ⊃ sdt[str][p0].meta = sdt0[str][p0].meta;sdt[str][p0].fid = sdt0[str][p0].fid))

Rule argument

Additional information

Used simplifier rules: entries-dom1: ⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt;%entries-dom2: ⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi;%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ str ∈ sdt → dnode(md, sdt)[str ' + p] = sdt[str][p]; (s)% ⊦ ⌜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)