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Proof step 42 from proof abs-nlinks-h (previous step)

tree[p0 + str '] = fnode(⌜ino1⌝ ), tree[p0] = dnode(md, sdt), ((m0 => mkdir(# entries0, md, m1, m2, entries0) * (abs(sdt -- str, entries0 -- str) * abs(sdt[str], entries0[str])) * abs|(tree, ino0, p0, m0)) )(dirs), ispath(p0, ino0, m0, dirs, files), str ∈ entries0, ino0 ∈ dirs, ino1 ∈ files, p0 ∈ tree, str ∈ sdt, ∀ alink. ¬ alink.pino ∈ dirs ∨ ¬ alink.name ∈ dirs[alink.pino].entries ∨ ino1 ≠ dirs[alink.pino].entries[alink.name], ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, entries0[str] ∈ dirs ∨ entries0[str] ∈ files ⊦ ino1 = entries0[str]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
⊦ p ∈ tree → tree[p + q] = tree[p][q]
Proof step 44:
⊦ abs(fnode(fid), ino) = (emp ∧ ⌜ino = fid.ino⌝ )

Rule argument

spec: directory-tree name: split-at seq: ⊦ p ∈ tree → tree[p + q] = tree[p][q] subst: [q, p, tree] → [str ', p0, tree] flags: <all paths>

Additional information

Used simplifier rules: abs-fnode: ⊦ abs(fnode(fid), ino) = (emp ∧ ⌜ino = fid.ino⌝ );%str ∈ sdt ⊦ dnode(md, sdt)[str '] = sdt[str]; (s)% ⊦ ⌜m⌝ .ino = m; (s)% ⊦ (⌜bv⌝ ∧ D) * D0 = (⌜bv⌝ ∧ D * D0); (s)% ⊦ emp * D = D; (s)%sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap]%and-ass: ⊦ ((D ∧ D0) ∧ D1) = (D ∧ D0 ∧ D1); from specification heap-sep[dir-heap]%and-com: ⊦ (D ∧ D0) = (D0 ∧ D); from specification heap-sep[dir-heap]% ⊦ ((D ∧ D0))(dirs) ↔ D(dirs) ∧ D0(dirs); (s)% ⊦ ⌜false⌝ = false; (s)% ⊦ ¬ false(dirs); (s) Used terms: str ', p0, tree