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Proof step 1 from proof fids-exists (conclusion)

⊦ ∃ fids. ∀ fid. fid ∈ fids ↔ fid ∈ tree.fids

User applied rule induction

And obtained the following new goals

Proof step 2:
∀ fid. Ptree(fnode(fid)), ∀ md, sdt. (∀ str. str ∈ sdt → Ptree(sdt[str])) → Ptree(dnode(md, sdt)) ⊦ Ptree(tree)
Proof step 3:
⊦ (∀ fid. ∀ tree. fnode(fid) = tree → (∃ fids. ∀ fid. fid ∈ fids ↔ fid ∈ tree.fids)) ∧ (∀ md, sdt. (∀ str. str ∈ sdt → (∀ tree. sdt[str] = tree → (∃ fids. ∀ fid. fid ∈ fids ↔ fid ∈ tree.fids))) → (∀ tree. dnode(md, sdt) = tree → (∃ fids. ∀ fid. fid ∈ fids ↔ fid ∈ tree.fids)))

Rule argument

spec: directory-tree name: tree-induction seq: ∀ fid. Ptree(fnode(fid)), ∀ md, sdt. (∀ str. str ∈ sdt → Ptree(sdt[str])) → Ptree(dnode(md, sdt)) ⊦ Ptree(tree) subst: [Ptree, tree] → [λ tree0. ∀ tree. tree0 = tree → (∃ fids. ∀ fid. fid ∈ fids ↔ fid ∈ tree.fids), tree]

Additional information

Used terms: λ tree0. ∀ tree. tree0 = tree → (∃ fids. ∀ fid. fid ∈ fids ↔ fid ∈ tree.fids), tree