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Proof step 734 from proof overwrite-write-page-clean (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 735:
⊦ n ∈ content1 → (content0 ⊕ content1)[n] = content1[n]
Proof step 736:
files[ino0] = mkfile(m4, md1, m5, content1), ((files ⊕ pcache) ⊕ icache)[ino] = mkfile(m0, md, m1, content), files[ino] = mkfile(m2, md0, m3, content0), ino ∈ files, ino0 ∈ files, ino ∈ icache, datakey(ino0, m) ∈ pcache, datakey(ino0, m) ∈ pcache[datakey(ino, m), mkpentry(false, buf)], m ∈ mkcontent(ino0, pcache[datakey(ino, m), mkpentry(false, buf)]) ⊦ (content1 ⊕ mkcontent(ino0, pcache))[m] = mkcontent(ino0, pcache[datakey(ino, m), mkpentry(false, buf)])[m], ino0 ∈ icache, m ∈ content

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-overwrite-content-old: ⊦ ¬ n ∈ content1 → (content0 ⊕ content1)[n] = content0[n];%in-mkcontent: ⊦ n ∈ mkcontent(ino, pcache) ↔ datakey(ino, n) ∈ pcache;% ⊦ key ∈ pcache[key0, pe] ↔ key = key0 ∨ key ∈ pcache; (ws) Used terms: m, mkcontent(ino0, pcache[datakey(ino, m), mkpentry(false, buf)]), content1