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Proof step 16 from proof abs-files-put (previous step)

ino ∈ files ⊦ abs-files(files)[⌜ino⌝ , mkfile(md, (content.stream to m).buf)] = abs-files(files[ino, mkfile(m, md, m0, content)])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ fs = fs0 ↔ (∀ fid. (fid ∈ fs ↔ fid ∈ fs0) ∧ (fid ∈ fs → fs[fid] = fs0[fid]))
Proof step 18:
ino ∈ files ⊦ ∀ fid. (fid ∈ abs-files(files)[⌜ino⌝ , mkfile(md, (content.stream to m).buf)] ↔ fid ∈ abs-files(files[ino, mkfile(m, md, m0, content)])) ∧ ( fid ∈ abs-files(files)[⌜ino⌝ , mkfile(md, (content.stream to m).buf)] → abs-files(files)[⌜ino⌝ , mkfile(md, (content.stream to m).buf)][fid] = abs-files(files[ino, mkfile(m, md, m0, content)])[fid])

Rule argument

spec: store, inst: file-data-store name: k-Extension seq: ⊦ fs = fs0 ↔ (∀ fid. (fid ∈ fs ↔ fid ∈ fs0) ∧ (fid ∈ fs → fs[fid] = fs0[fid])) subst: [fs0, fs] → [abs-files(files[ino, mkfile(m, md, m0, content)]), abs-files(files) [⌜ino⌝ , mkfile(md, (content.stream to m).buf) ]] flags: <all paths>

Additional information

Used terms: abs-files(files[ino, mkfile(m, md, m0, content)]), abs-files(files)[⌜ino⌝ , mkfile(md, (content.stream to m).buf)]