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

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Proof step 5 from proof abs-tree (previous step)

((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 ⊦ (p ∈ tree0 ↔ p ∈ tree1) ∧ (p ∈ tree0 → (tree0[p].dir? ↔ tree1[p].dir?) ∧ (tree0[p].dir? ⊃ tree0[p].meta = tree1[p].meta;tree0[p].fid = tree1[p].fid))

User applied rule structural induction

And obtained the following new goals

Proof step 6:
((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 ⊦ (ε ∈ tree0 ↔ ε ∈ tree1) ∧ (ε ∈ tree0 → (tree0[ε].dir? ↔ tree1[ε].dir?) ∧ (tree0[ε].dir? ⊃ tree0[ε].meta = tree1[ε].meta;tree0[ε].fid = tree1[ε].fid))
Proof step 32:
((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 + p0 ∈ tree0 ↔ str + p0 ∈ tree1) ∧ ( str + p0 ∈ tree0 → (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

indvar: p

Additional information

Used simplifier rules: path freely generated by ε, +;;