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Proof step 15 from proof absc-drop (previous step)

dnode(md, sdt)[parent(str0 ' + p0)].dir?, parent(str0 ' + p0) ∈ dnode(md, sdt), q0 ⊑ p0, ∀ tree, p, tree0, ino0, ino1, D, dirs. tree[parent(p)].dir? ∧ parent(p) ∈ tree ∧ q0 ⊑ p → (((abs|(tree[p, tree0], ino0, q0, ino1) * D))(dirs) ↔ ((abs|(tree, ino0, q0, ino1) * D))(dirs)) ∨ p = ε ⊦ ((abs|(dnode(md, sdt)[str0 ' + p0, tree0], ino0, str0 ' + q0, ino1) * D))(dirs) ↔ (∃ ad. str0 ∈ ad.entries ∧ ad.meta = md ∧ ad.size = # ad.entries ∧ ((((abs|(sdt[str0], ad.entries[str0], q0, ino1) * abs(sdt -- str0, ad.entries -- str0)) * ino0 => ad) * D))(dirs))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, inst: path name: bapp seq: ⊦ p ≠ ε → parent(p0 + p) = p0 + parent(p) subst: [p, p0] → [p0, str0 '] flags: <all paths>

Additional information

Used simplifier rules: absc-dnode-rec: ⊦ abs|(dnode(md, sdt), ino, str ' + p, ino0) = _∃ (λ ad. ino => ad * abs|(sdt, ad.entries, str ' + p, ino0) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ );%absc-entries: ⊦ abs|(sdt, entries, str ' + p, ino0) = ( ⌜str ∈ entries⌝ ∧ abs|(sdt[str], entries[str], p, ino0) * abs(sdt -- str, entries -- str));%absc-emp: ⊦ abs|(tree, ino, ε, ino0) = (emp ∧ ⌜ino = ino0⌝ );% ⊦ p + ε = p; (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ dnode(md, sdt)[str ', tree0] = dnode(md, sdt[str, tree0]); (s)% ⊦ sdt[str, tree][str] = tree; (s)% ⊦ sdt[str, tree] -- str = sdt -- str; (s)% ⊦ (⌜bv⌝ ∧ D) * D0 = (⌜bv⌝ ∧ D * D0); (s)%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]%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]% ⊦ ((_∃ adD * D))(dirs) ↔ (∃ ad. ((adD(ad) * D))(dirs)); (s)% ⊦ ((D ∧ D0))(dirs) ↔ D(dirs) ∧ D0(dirs); (s)% ⊦ ⌜bv⌝ (dirs) ↔ bv; (s)% ⊦ p ⊑ ε ↔ p = ε; (s)% ⊦ parent(str ') = ε; (s)% ⊦ ε ∈ tree; (s)% ⊦ tree[ε] = tree; (s)% ⊦ dnode(md, sdt).dir?; (s)% ⊦ emp * D = D; (s) Used terms: p0, str0 '