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Proof step 1 from proof crash-exists-h (conclusion)

⊦ ∃ fs1. (∀ fid. fid ∈ fs1 ↔ fid ∈ tree.fids ∧ fid ∈ fs0) ∧ (∀ fid. fid ∈ fs1 → fs1[fid] = fs0[fid])

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, fs0. fnode(fid) = tree → (∃ fs1. (∀ fid. fid ∈ fs1 ↔ fid ∈ tree.fids ∧ fid ∈ fs0) ∧ (∀ fid. fid ∈ fs1 → fs1[fid] = fs0[fid]))) ∧ (∀ md, sdt. (∀ str. str ∈ sdt → (∀ tree, fs0. sdt[str] = tree → (∃ fs1. (∀ fid. fid ∈ fs1 ↔ fid ∈ tree.fids ∧ fid ∈ fs0) ∧ (∀ fid. fid ∈ fs1 → fs1[fid] = fs0[fid])))) → (∀ tree, fs0. dnode(md, sdt) = tree → (∃ fs1. (∀ fid. fid ∈ fs1 ↔ fid ∈ tree.fids ∧ fid ∈ fs0) ∧ (∀ fid. fid ∈ fs1 → fs1[fid] = fs0[fid]))))

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, fs0. tree0 = tree → (∃ fs1. (∀ fid. fid ∈ fs1 ↔ fid ∈ tree.fids ∧ fid ∈ fs0) ∧ (∀ fid. fid ∈ fs1 → fs1[fid] = fs0[fid])), tree]

Additional information

Used terms: λ tree0. ∀ tree, fs0. tree0 = tree → (∃ fs1. (∀ fid. fid ∈ fs1 ↔ fid ∈ tree.fids ∧ fid ∈ fs0) ∧ (∀ fid. fid ∈ fs1 → fs1[fid] = fs0[fid])), tree