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Proof step 3 from proof absc-dnode-rec (previous step)

⊦ abs|(dnode(md, sdt), ino, str ' + p, ino0) = _∃ (λ ad. (⌜str ∈ ad.entries⌝ ∧ (abs|(sdt[str], ad.entries[str], p, ino0) * abs(sdt -- str, ad.entries -- str)) * ino => ad) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ )

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ p ≠ ε → abs|(dnode(md, sdt), ino, p, ino0) = _∃(λ ad. ino => ad * abs|(sdt, ad.entries, p, ino0) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ )
Proof step 5:
⊦ abs|(sdt, entries, str ' + p, ino0) = (⌜str ∈ entries⌝ ∧ abs|(sdt[str], entries[str], p, ino0) * abs(sdt -- str, entries -- str))

Rule argument

spec: <toplevel> name: absc-dnode seq: ⊦ p ≠ ε → abs|(dnode(md, sdt), ino, p, ino0) = _∃ (λ ad. ino => ad * abs|(sdt, ad.entries, p, ino0) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ ) subst: [ino0, p, ino, sdt, md] → [ino0, str ' + p, ino, sdt, md] flags: <all paths>

Additional information

Used simplifier rules: absc-entries: ⊦ abs|(sdt, entries, str ' + p, ino0) = ( ⌜str ∈ entries⌝ ∧ abs|(sdt[str], entries[str], p, ino0) * abs(sdt -- str, entries -- str));%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ (⌜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]% ⊦ p + p0 = ε ↔ p = ε ∧ p0 = ε; (as)% ⊦ str ' ≠ ε; (s) Used terms: ino0, str ' + p, ino, sdt, md