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Proof step 20 from proof in-put (previous step)

sdt0[str0][p0] = dnode(md, sdt), q = str1 + q0, p0 ∈ sdt0[str0], str0 ∈ sdt0, ∀ 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))) ⊦ q ∈ dnode(md0, sdt0)[str0 ' + p0, dnode(md, sdt[str, tree0])] ↔ ((str0 ' + p0) + str ' ⊑ q ⊃ restn(# p0 + 1, q.tail) ∈ tree0;q ∈ dnode(md0, sdt0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
⊦ p + q ∈ tree ↔ p ∈ tree ∧ q ∈ tree[p]
Proof step 22:
p ≠ ε ⊦ restn(n + 1, p) = restn(n, p.tail)
Proof step 23:
⊦ str ' + p ∈ dnode(md, sdt) ↔ str ∈ sdt ∧ p ∈ sdt[str]
Proof step 24:
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 ' ∈ dnode(md0, sdt0)[str0 ' + p0, dnode(md, sdt[str, tree0])] ∧ q0 ∈ dnode(md0, sdt0)[str0 ' + p0, dnode(md, sdt[str, tree0])][str1 '] ↔ ((str0 ' + p0) + str ' ⊑ str1 ' + q0 ⊃ restn(# p0, q0.tail) ∈ tree0;str1 ∈ sdt0 ∧ q0 ∈ sdt0[str1])

Rule argument

Additional information

Used simplifier rules: split-in: ⊦ p + q ∈ tree ↔ p ∈ tree ∧ q ∈ tree[p];%one-restn: p ≠ ε ⊦ restn(n + 1, p) = restn(n, p.tail);%in-dnode: ⊦ str ' + p ∈ dnode(md, sdt) ↔ str ∈ sdt ∧ p ∈ sdt[str];%c: ⊦ m + n = n + m; from specification nat-basic[directory-tree-inst]%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic[directory-tree-inst]% ⊦ p + str ' ≠ ε; (s)% ⊦ str ' = str0 ' + p ↔ str = str0 ∧ p = ε; (ws)% ⊦ str ' + p ≠ ε; (s)% ⊦ p ⊑ str ' ↔ p = ε ∨ p = str '; (s)% ⊦ p + ε = p; (s)% ⊦ (str ' + p).tail = p; (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ str + p = str ' + p; (s)