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Proof step 13 from proof fids-in (previous step)

fid ∈ sdt.fids, ∀ str. str ∈ sdt → (∀ fid. fid ∈ sdt[str].fids → (∃ p. p ∈ sdt[str] ∧ sdt[str][p] = fnode(fid))), ∀ sdt0, fid, md. # sdt0 < # sdt → fid ∈ sdt0.fids ∧ (∀ str. str ∈ sdt0 → (∀ fid. fid ∈ sdt0[str].fids → (∃ p. p ∈ sdt0[str] ∧ sdt0[str][p] = fnode(fid)))) → (∃ p. p ∈ dnode(md, sdt0) ∧ dnode(md, sdt0)[p] = fnode(fid)) ⊦ ∀ str, tree, sdt0. ¬(sdt = sdt0[str, tree] ∧ ¬ str ∈ sdt0), ∃ p. p ∈ dnode(md, sdt) ∧ dnode(md, sdt)[p] = fnode(fid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
⊦ ¬ str ∈ sdt → sdt[str, tree].fids = sdt.fids ∪ tree.fids
Proof step 15:
fid ∈ sdt0.fids ∪ tree.fids, ∀ str0. str0 ∈ sdt0[str, tree] → (∀ fid. fid ∈ sdt0[str, tree][str0].fids → (∃ p. p ∈ sdt0[str, tree][str0] ∧ sdt0[str, tree][str0][p] = fnode(fid))), ∀ sdt1, fid, md. # sdt1 < # sdt0[str, tree] → fid ∈ sdt1.fids ∧ (∀ str. str ∈ sdt1 → (∀ fid. fid ∈ sdt1[str].fids → (∃ p. p ∈ sdt1[str] ∧ sdt1[str][p] = fnode(fid)))) → (∃ p. p ∈ dnode(md, sdt1) ∧ dnode(md, sdt1)[p] = fnode(fid)) ⊦ str ∈ sdt0, ∃ p. p ∈ dnode(md, sdt0[str, tree]) ∧ dnode(md, sdt0[str, tree])[p] = fnode(fid)

Rule argument

Additional information

Used simplifier rules: fids-ins: ⊦ ¬ str ∈ sdt → sdt[str, tree].fids = sdt.fids ∪ tree.fids;