[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 6 from proof absc-drop (previous step)

tree[parent(p)].dir?, parent(p) ∈ tree, q ⊑ p ⊦ ((abs|(tree[p, tree0], ino0, q, ino1) * D))(dirs) ↔ ((abs|(tree, ino0, q, ino1) * D))(dirs), p = ε

User applied rule structural induction

And obtained the following new goals

Proof step 7:
⊦ abs|(tree, ino, ε, ino0) = (emp ∧ ⌜ino = ino0⌝ )
Proof step 8:
tree[parent(p)].dir?, parent(p) ∈ tree, str + q0 ⊑ p, ∀ 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|(tree[p, tree0], ino0, str + q0, ino1) * D))(dirs) ↔ ((abs|(tree, ino0, str + q0, ino1) * D))(dirs), p = ε

Rule argument

indvar: q

Additional information

Used simplifier rules: absc-emp: ⊦ abs|(tree, ino, ε, ino0) = (emp ∧ ⌜ino = ino0⌝ );%path freely generated by ε, +;;% ⊦ (⌜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)% ⊦ ⌜true⌝ = true; (s)% ⊦ (true ∧ D) = D; (s)