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Proof step 1 from proof entries-unfold (conclusion)

str ∈ sdt ⊦ ((abs(sdt, sdi) * D))(dirs) ↔ (∃ tree, ino, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 2:
⊦ ¬ name ∈ sdt → abs(sdt[name, tree], entries) = (⌜name ∈ entries⌝ ∧ abs(tree, entries[name]) * abs(sdt, entries -- name))
Proof step 3:
⊦ ((abs(sdt, sdi) * D))(dirs) ∧ ¬ str ∈ sdt → sdi -- str = sdi
Proof step 4:
⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi
Proof step 5:
⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt
Proof step 6:
((abs(sdt, sdi) * D))(dirs), str ∈ sdi, str ∈ sdt ⊦ ∃ tree, ino, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs)

Rule argument

Additional information

Used simplifier rules: abs-rec: ⊦ ¬ name ∈ sdt → abs(sdt[name, tree], entries) = ( ⌜name ∈ entries⌝ ∧ abs(tree, entries[name]) * abs(sdt, entries -- name));%entries-del: ⊦ ((abs(sdt, sdi) * D))(dirs) ∧ ¬ str ∈ sdt → sdi -- str = sdi;%entries-dom2: ⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi;%entries-dom1: ⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt;%and-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]% ⊦ (true ∧ D) = D; (s)%sep-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]% ⊦ entries[str, ino] -- str = entries -- str; (s)% ⊦ entries[str, ino][str] = ino; (s)% ⊦ ⌜true⌝ = true; (s)% ⊦ str ∈ entries[str, ino]; (s)