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Proof step 1 from proof abs-files-orphans (conclusion)

⊦ abs-files(fs, ri \ orphans(fs, ri)) = abs-files(fs, ri) \ abs-keys(orphans(fs, ri))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 2:
⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino]))
Proof step 3:
⊦ ∀ ino. (ino ∈ abs-files(fs, ri \ orphans(fs, ri)) ↔ ino ∈ abs-files(fs, ri) \ abs-keys(orphans(fs, ri))) ∧ (ino ∈ abs-files(fs, ri \ orphans(fs, ri)) → abs-files(fs, ri \ orphans(fs, ri))[ino] = (abs-files(fs, ri) \ abs-keys(orphans(fs, ri)))[ino])

Rule argument

spec: heap, inst: file-heap name: ext seq: ⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino])) subst: [files0, files] → [abs-files(fs, ri) \ abs-keys(orphans(fs, ri)), abs-files(fs, ri \ orphans(fs, ri))] flags: <all paths>

Additional information

Used terms: abs-files(fs, ri) \ abs-keys(orphans(fs, ri)), abs-files(fs, ri \ orphans(fs, ri))