[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 4 from proof entries-dom (previous step)

((abs(sdt, sdi) * D))(dirs), ∀ sdt0, sdi, D, dirs, str. # sdt0 < # sdt → ((abs(sdt0, sdi) * D))(dirs) → (str ∈ sdi ↔ str ∈ sdt0) ⊦ str ∈ sdi ↔ str ∈ sdt

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
sdt ≠ ∅ ⊦ ((abs(sdt, sdi) * D))(dirs) ↔ (∃ tree, ino, str, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs))
Proof step 6:
⊦ abs(∅, entries) = (⌜entries = ∅⌝ ∧ emp)
Proof step 7:
∃ tree, ino, str, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs), ∀ sdt0, sdi, D, dirs, str. # sdt0 < # sdt → ((abs(sdt0, sdi) * D))(dirs) → (str ∈ sdi ↔ str ∈ sdt0) ⊦ str ∈ sdi ↔ str ∈ sdt, sdt = ∅

Rule argument

spec: <toplevel> name: entries-any seq: sdt ≠ ∅ ⊦ ((abs(sdt, sdi) * D))(dirs) ↔ (∃ tree, ino, str, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs)) subst: [dirs, D, sdi, sdt] → [dirs, D, sdi, sdt] flags: <all paths>

Additional information

Used simplifier rules: abs-empty: ⊦ abs(∅, entries) = (⌜entries = ∅⌝ ∧ emp);% ⊦ ¬ str ∈ ∅; (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)% ⊦ ¬ str ∈ ∅; (s) Used terms: dirs, D, sdi, sdt