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Proof step 113 from proof refine-truncate-buff-some (previous step)

# buf-opt0.buf = VFS_PAGE_SIZE, VFS_PAGE_SIZE = n0 + m2 % VFS_PAGE_SIZE, ((cache_files ⊕ cache_pcache0) ⊕ cache_icache)[m3] = mkfile(m0, md0, # links(m3, cache_dirs), content0), cache_files[m3] = mkfile(m1, md2, # links(m3, cache_dirs), content1), afs_files = ((cache_files ⊕ cache_pcache0) ⊕ cache_icache )[m3, mkfile (m0, md0, # links(m3, cache_dirs), content0[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, VFS_PAGE_SIZE - m2 % VFS_PAGE_SIZE)] upto m2) ], cache_dirs2 = cache_dirs, m2 % VFS_PAGE_SIZE ≠ 0, buf-opt0.some?, afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache0) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache0, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache0), m3 ∈ cache_files, m2 ≤ m, cache_sync → synced(cache_files, cache_icache, cache_pcache0), ∀ key. key ∈ cache_pcache0 → key.data? ⊦ ⟨afs_inode_sync#(mkinode(m3, md, false, # links(m3, cache_dirs), 0, m); cache_dirs, afs_files); inode := (m3 ∈ cache_dirs ⊃ mkinode(m3, cache_dirs[m3].meta, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size) ;mkinode(m3, afs_files[m3].meta, false, afs_files[m3].nlink, 0, afs_files[m3].size) ); err := ESUCCESS; asynced := ?⟩ ( cache_dirs = cache_dirs2 ∧ afs_files = ( cache_files [m3, mkfile(m, md, # links(m3, cache_dirs2), content1[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2) ] ⊕ truncate(m3, m2, cache_pcache0)[datakey(m3, m2 / VFS_PAGE_SIZE), mkpentry(false, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0))]) ⊕ cache_icache[m3, mkientry(false, mkinode(m3, md, false, # links(m3, cache_dirs2), 0, m))] ∧ (asynced ↔ asynced1) ∧ fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0) = fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, VFS_PAGE_SIZE - m2 % VFS_PAGE_SIZE) ∧ (m3 ∈ cache_dirs2 ⊃ mkinode(m3, cache_dirs2[m3].meta, true, cache_dirs2[m3].nlink, cache_dirs2[m3].nsubdirs, cache_dirs2[m3].size) ;mkinode(m3, md, false, # links(m3, cache_dirs2), 0, m) ) = inode ∧ ESUCCESS = err), ⟨fail#(; ; err); asynced := ?⟩ ( (cache_files ⊕ cache_pcache0) ⊕ cache_icache = ( cache_files [m3, mkfile(m, md, # links(m3, cache_dirs), content1[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2) ] ⊕ truncate(m3, m2, cache_pcache0)[datakey(m3, m2 / VFS_PAGE_SIZE), mkpentry(false, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0))]) ⊕ cache_icache[m3, mkientry(false, mkinode(m3, md, false, # links(m3, cache_dirs), 0, m))] ∧ (asynced ↔ asynced1) ∧ some(fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0)) = buf-opt0 ∧ mkinode(m3, md, false, # links(m3, cache_dirs), 0, m) = mkinode(m3, md, false, # links(m3, cache_dirs), 0, m2) ∧ asynced2 ∧ ESUCCESS = err)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 114:
# buf-opt0.buf = VFS_PAGE_SIZE, VFS_PAGE_SIZE = n0 + m2 % VFS_PAGE_SIZE, VFS_PAGE_SIZE = n1 + m2 % VFS_PAGE_SIZE, cache_files[m3] = mkfile(m1, md2, # links(m3, cache_dirs), content1), ((cache_files ⊕ cache_pcache0) ⊕ cache_icache)[m3] = mkfile(m0, md0, # links(m3, cache_dirs), content0), afs_files = ((cache_files ⊕ cache_pcache0) ⊕ cache_icache )[m3, mkfile(m0, md0, # links(m3, cache_dirs), content0[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1)] upto m2) ], cache_dirs2 = cache_dirs, buf-opt0.some?, afs-cons(cache_dirs, (cache_files ⊕ cache_pcache0) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache0, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache0), m3 ∈ cache_files, m2 ≤ m, m2 % VFS_PAGE_SIZE ≤ VFS_PAGE_SIZE, cache_sync → synced(cache_files, cache_icache, cache_pcache0), ∀ key. key ∈ cache_pcache0 → key.data? ⊦ ⟨afs_inode_sync#(mkinode(m3, md, false, # links(m3, cache_dirs), 0, m); cache_dirs, afs_files); inode := (m3 ∈ cache_dirs ⊃ mkinode(m3, cache_dirs[m3].meta, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size) ;mkinode(m3, afs_files[m3].meta, false, afs_files[m3].nlink, 0, afs_files[m3].size) ); err := ESUCCESS; asynced := ?⟩ ( cache_dirs = cache_dirs2 ∧ afs_files = ( cache_files [m3, mkfile(m, md, # links(m3, cache_dirs2), content1[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2) ] ⊕ truncate(m3, m2, cache_pcache0)[datakey(m3, m2 / VFS_PAGE_SIZE), mkpentry(false, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0))]) ⊕ cache_icache[m3, mkientry(false, mkinode(m3, md, false, # links(m3, cache_dirs2), 0, m))] ∧ (asynced ↔ asynced1) ∧ fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0) = fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1) ∧ (m3 ∈ cache_dirs2 ⊃ mkinode(m3, cache_dirs2[m3].meta, true, cache_dirs2[m3].nlink, cache_dirs2[m3].nsubdirs, cache_dirs2[m3].size) ;mkinode(m3, md, false, # links(m3, cache_dirs2), 0, m) ) = inode ∧ ESUCCESS = err), ⟨fail#(; ; err); asynced := ?⟩ ( (cache_files ⊕ cache_pcache0) ⊕ cache_icache = ( cache_files [m3, mkfile(m, md, # links(m3, cache_dirs), content1[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2) ] ⊕ truncate(m3, m2, cache_pcache0)[datakey(m3, m2 / VFS_PAGE_SIZE), mkpentry(false, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0))]) ⊕ cache_icache[m3, mkientry(false, mkinode(m3, md, false, # links(m3, cache_dirs), 0, m))] ∧ (asynced ↔ asynced1) ∧ some(fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n0)) = buf-opt0 ∧ mkinode(m3, md, false, # links(m3, cache_dirs), 0, m) = mkinode(m3, md, false, # links(m3, cache_dirs), 0, m2) ∧ asynced2 ∧ ESUCCESS = err), m2 % VFS_PAGE_SIZE = 0

Rule argument

spec: nat name: elim-sub seq: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n subst: [n, m, n0] → [m2 % VFS_PAGE_SIZE, VFS_PAGE_SIZE, n]

Additional information

Used simplifier rules: ⊦ VFS_PAGE_SIZE ≠ 0; (s)%n ≠ 0 ⊦ m % n ≤ n; (s)%n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: m2 % VFS_PAGE_SIZE, VFS_PAGE_SIZE, n