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Proof step 20 from proof abs-fold-del-p (previous step)

# entries = m + 1, tree[p] = dnode(md, sdt), ((abs|(tree, ROOT_INO, p, ino0) * ino0 => mkdir(m, md, m0, m1, entries -- str) * abs(sdt -- str, entries -- str) * D))(dirs), str ∈ entries, p ∈ tree ⊦ entries = ∅, (((ino0 => mkdir(m, md, m0, m1, entries -- str) * abs(sdt -- str, entries -- str) * abs|(tree[p, dnode(md, sdt -- str)], ROOT_INO, p, ino0)) * D) )(dirs), p ≠ ε ∧ parent(p) ∈ tree ∧ tree[parent(p)].dir?

User applied rule insert elim lemma

And closed the goal!

Rule argument

spec: path name: parent-elim seq: p ≠ ε ⊦ q = parent(p) ∧ str = p.last ↔ p = q + str ' subst: [p, q, str] → [p, q, str0]

Additional information

Used simplifier rules: ⊦ p + str ' ≠ ε; (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ p + q ∈ tree ↔ p ∈ tree ∧ q ∈ tree[p]; (s)% ⊦ ¬ tree.dir? → (p ∈ tree ↔ p = ε); (s)% ⊦ str ' ≠ ε; (s)% ⊦ tree[ε] = tree; (s)% ⊦ ε ∈ tree; (s)% ⊦ abs|(tree, ino, ε, ino0) = (emp ∧ ⌜ino = ino0⌝ ); (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)% ⊦ ⌜bv⌝ (dirs) ↔ bv; (s)% ⊦ tree[ε, tree0] = tree0; (s)% ⊦ ⌜true⌝ = true; (s)% ⊦ (true ∧ D) = D; (s)%p ≠ ε ⊦ q = parent(p) ∧ str = p.last ↔ p = q + str '; (elim) Used terms: p, q, str0