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Proof step 129 from proof overwrite-write-page-clean (previous step)

icache[ino] = mkientry(boolvar, mkinode(m0, md, false, m1, m4, m5)), files[ino] = mkfile(m2, md0, m3, content0), n ∈ content0[m, buf], ino ∈ files, ino ∈ icache ⊦ (content0 ⊕ mkcontent(ino, pcache))[m, buf][n] = (content0[m, buf] ⊕ mkcontent(ino, pcache[datakey(ino, m), mkpentry(false, buf)]))[n], datakey(ino, n) = datakey(ino, m), datakey(ino, n) ∈ pcache

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 130:
⊦ n ∈ content1 → (content0 ⊕ content1)[n] = content1[n]
Proof step 131:
files[ino] = mkfile(m2, md0, m3, content0), icache[ino] = mkientry(boolvar, mkinode(m0, md, false, m1, m4, m5)), ino ∈ icache, ino ∈ files, n ∈ content0[m, buf] ⊦ (content0 ⊕ mkcontent(ino, pcache))[m, buf][n] = (content0[m, buf] ⊕ mkcontent(ino, pcache[datakey(ino, m), mkpentry(false, buf)]))[n], datakey(ino, n) = datakey(ino, m), datakey(ino, n) ∈ pcache, n ∈ mkcontent(ino, 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] → [n, mkcontent (ino, pcache[datakey(ino, m), mkpentry(false, buf)]), content0[m, buf]] 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: n, mkcontent(ino, pcache[datakey(ino, m), mkpentry(false, buf)]), content0[m, buf]