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

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Proof step 33 from proof abs-tree (previous step)

((abs(tree1, ino) * D1))(dirs), ((abs(tree0, ino) * D0))(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)), ∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files, ∀ ino. ⌜ino⌝ ∈ tree0.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs ⊦ (str ' ∈ tree0 ∧ p0 ∈ tree0[str '] ↔ str ' ∈ tree1 ∧ p0 ∈ tree1[str ']) ∧ ( str ' ∈ tree0 ∧ p0 ∈ tree0[str '] → (tree0[str ' + p0].dir? ↔ tree1[str ' + p0].dir?) ∧ (tree0[str ' + p0].dir? ⊃ tree0[str ' + p0].meta = tree1[str ' + p0].meta;tree0[str ' + p0].fid = tree1[str ' + p0].fid))

User applied rule constructor cut

And obtained the following new goals

Proof step 34:
tree0 = dnode(md, sdt), ((abs(tree0, ino) * D0))(dirs), ((abs(tree1, ino) * D1))(dirs), ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree0.fids → ino ∈ files, ∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files, ∀ 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)) ⊦ (str ' ∈ tree0 ∧ p0 ∈ tree0[str '] ↔ str ' ∈ tree1 ∧ p0 ∈ tree1[str ']) ∧ ( str ' ∈ tree0 ∧ p0 ∈ tree0[str '] → (tree0[str ' + p0].dir? ↔ tree1[str ' + p0].dir?) ∧ (tree0[str ' + p0].dir? ⊃ tree0[str ' + p0].meta = tree1[str ' + p0].meta;tree0[str ' + p0].fid = tree1[str ' + p0].fid))
Proof step 67:
tree0 = fnode(fid), ((abs(tree0, ino) * D0))(dirs), ((abs(tree1, ino) * D1))(dirs), ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree0.fids → ino ∈ files, ∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files, ∀ 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)) ⊦ (str ' ∈ tree0 ∧ p0 ∈ tree0[str '] ↔ str ' ∈ tree1 ∧ p0 ∈ tree1[str ']) ∧ ( str ' ∈ tree0 ∧ p0 ∈ tree0[str '] → (tree0[str ' + p0].dir? ↔ tree1[str ' + p0].dir?) ∧ (tree0[str ' + p0].dir? ⊃ tree0[str ' + p0].meta = tree1[str ' + p0].meta;tree0[str ' + p0].fid = tree1[str ' + p0].fid))

Rule argument

var: tree0

Additional information

Used simplifier rules: directory-tree, entries-tree generated by ∅, dnode, fnode, [ ];;