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Proof step 50 from proof in-put (previous step)

sdt0[str0][p0] = dnode(md, sdt), str0 ∈ sdt0, p0 ∈ sdt0[str0], ∀ md0, sdt0, md, sdt, q, str, tree0. dnode(md0, sdt0)[p0] = dnode(md, sdt) ∧ p0 ∈ dnode(md0, sdt0) → (q ∈ dnode(md0, sdt0)[p0, dnode(md, sdt[str, tree0])] ↔ (p0 + str ' ⊑ q ⊃ restn(# p0, q.tail) ∈ tree0;q ∈ dnode(md0, sdt0))) ⊦ str1 ∈ sdt0[str0, sdt0[str0][p0, dnode(md, sdt[str, tree0])]] ∧ q0 ∈ dnode(md0, sdt0[str0, sdt0[str0][p0, dnode(md, sdt[str, tree0])]])[str1 '] ↔ ((str0 ' + p0) + str ' ⊑ str1 ' + q0 ⊃ restn(# p0, q0.tail) ∈ tree0;str1 ∈ sdt0 ∧ q0 ∈ sdt0[str1]), p0 = ε

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 51:
⊦ str ∈ sdt ↔ (∃ tree, sdt0. sdt = sdt0[str, tree] ∧ ¬ str ∈ sdt0)
Proof step 52:
sdt0[str0][p0] = dnode(md, sdt), p0 ∈ sdt0[str0], ∀ md0, sdt0, md, sdt, q, str, tree0. dnode(md0, sdt0)[p0] = dnode(md, sdt) ∧ p0 ∈ dnode(md0, sdt0) → (q ∈ dnode(md0, sdt0)[p0, dnode(md, sdt[str, tree0])] ↔ (p0 + str ' ⊑ q ⊃ restn(# p0, q.tail) ∈ tree0;q ∈ dnode(md0, sdt0))), ∃ tree, sdt1. sdt0 = sdt1[str0, tree] ∧ ¬ str0 ∈ sdt1 ⊦ str1 ∈ sdt0[str0, sdt0[str0][p0, dnode(md, sdt[str, tree0])]] ∧ q0 ∈ dnode(md0, sdt0[str0, sdt0[str0][p0, dnode(md, sdt[str, tree0])]])[str1 '] ↔ ((str0 ' + p0) + str ' ⊑ str1 ' + q0 ⊃ restn(# p0, q0.tail) ∈ tree0;str1 ∈ sdt0 ∧ q0 ∈ sdt0[str1]), p0 = ε

Rule argument

spec: store, inst: directory-tree-inst name: exin seq: ⊦ str ∈ sdt ↔ (∃ tree, sdt0. sdt = sdt0[str, tree] ∧ ¬ str ∈ sdt0) subst: [sdt, str] → [sdt0, str0] flags: <all paths>

Additional information

Used terms: sdt0, str0