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Proof step 330 from proof overwrite-write-page-clean (previous step)

files[ino0] = mkfile(m8, md2, m9, content1), icache[ino0] = mkientry(boolvar, mkinode(m4, md1, false, m5, m6, m7)), files[ino] = mkfile(m2, md0, m3, content0), ((files ⊕ pcache) ⊕ icache)[ino] = mkfile(m0, md, m1, content), m ∈ content1, ino0 ∈ icache, ino ∈ icache, ino0 ∈ files, ino ∈ files ⊦ (content1 ⊕ mkcontent(ino0, pcache))[m] = (content1 ⊕ mkcontent(ino0, pcache[datakey(ino, m), mkpentry(false, buf)]))[m], datakey(ino0, m) = datakey(ino, m), datakey(ino0, m) ∈ pcache

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 331:
⊦ n ∈ content1 → (content0 ⊕ content1)[n] = content1[n]
Proof step 332:
((files ⊕ pcache) ⊕ icache)[ino] = mkfile(m0, md, m1, content), files[ino] = mkfile(m2, md0, m3, content0), icache[ino0] = mkientry(boolvar, mkinode(m4, md1, false, m5, m6, m7)), files[ino0] = mkfile(m8, md2, m9, content1), ino ∈ files, ino0 ∈ files, ino ∈ icache, ino0 ∈ icache, m ∈ content1 ⊦ (content1 ⊕ mkcontent(ino0, pcache))[m] = (content1 ⊕ mkcontent(ino0, pcache[datakey(ino, m), mkpentry(false, buf)]))[m], datakey(ino0, m) = datakey(ino, m), datakey(ino0, m) ∈ pcache, m ∈ mkcontent(ino0, pcache[datakey(ino, m), mkpentry(false, buf)])

Rule argument

spec: <toplevel> name: at-overwrite-content-new seq: ⊦ n ∈ content1 → (content0 ⊕ content1)[n] = content1[n] subst: [n, content1, content0] → [m, mkcontent (ino0, pcache[datakey(ino, m), mkpentry(false, buf)]), content1] flags: <all paths>

Additional information

Used simplifier rules: at-mkcontent: ⊦ mkcontent(ino, pcache)[n] = pcache[datakey(ino, n)].page;%in-mkcontent: ⊦ n ∈ mkcontent(ino, pcache) ↔ datakey(ino, n) ∈ pcache;% ⊦ key ≠ key0 → pcache[key0, pe][key] = pcache[key]; (s)% ⊦ ¬ key ∈ pcache → pcache[key] = (∅)[key]; (s)% ⊦ key ≠ key0 → (key ∈ pcache[key0, pe] ↔ key ∈ pcache); (s) Used terms: m, mkcontent(ino0, pcache[datakey(ino, m), mkpentry(false, buf)]), content1