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Proof step 72 from proof inv-fsync (previous step)

files1[m0] = mkfile(m4, md, # links(m0, dirs0), content2), files0[m] = mkfile(m3, md2, # links(m, dirs1), content1), files2[m] = mkfile(m1, md1, # links(m, dirs1), content0), afs-cons(dirs0, files1), afs-cons(dirs1, files0), afs-cons(dirs1, files2), pcache-cons(pcache, dirs1, files2), synced(dirs0, icache0, pcache0), synced(dirs1, icache, pcache0), synced(dirs1, icache, pcache), m0 ∈ files1, m ∈ files2, m ∈ files0, ∀ ino. ino ∈ icache0 → icache0[ino].inode.ino = ino ∧ (icache0[ino].dirty ⊃ icache0[ino].inode.ino ∈ files1 ∧ icache0[ino].inode.nlink = files1[icache0[ino].inode.ino].nlink ∧ icache0[ino].inode.nsubdirs = 0 ∧ ¬ icache0[ino].inode.directory ; (ino ∈ dirs0 ∨ ino ∈ files1) ∧ icache0[ino].inode = (icache0[ino].inode.ino ∈ dirs0 ⊃ mkinode (icache0[ino].inode.ino, dirs0[icache0[ino].inode.ino].meta, true, dirs0[icache0[ino].inode.ino].nlink, dirs0[icache0[ino].inode.ino].nsubdirs, dirs0[icache0[ino].inode.ino].size) ;mkinode (icache0[ino].inode.ino, files1[icache0[ino].inode.ino].meta, false, files1[icache0[ino].inode.ino].nlink, 0, files1[icache0[ino].inode.ino].size) ) ), ∀ ino. ino ∈ icache → icache[ino].inode.ino = ino ∧ (icache[ino].dirty ⊃ icache[ino].inode.ino ∈ files0 ∧ icache[ino].inode.nlink = files0[icache[ino].inode.ino].nlink ∧ icache[ino].inode.nsubdirs = 0 ∧ ¬ icache[ino].inode.directory ; (ino ∈ dirs1 ∨ ino ∈ files0) ∧ icache[ino].inode = (icache[ino].inode.ino ∈ dirs1 ⊃ mkinode (icache[ino].inode.ino, dirs1[icache[ino].inode.ino].meta, true, dirs1[icache[ino].inode.ino].nlink, dirs1[icache[ino].inode.ino].nsubdirs, dirs1[icache[ino].inode.ino].size) ;mkinode (icache[ino].inode.ino, files0[icache[ino].inode.ino].meta, false, files0[icache[ino].inode.ino].nlink, 0, files0[icache[ino].inode.ino].size) ) ), ∀ ino. ino ∈ icache → icache[ino].inode.ino = ino ∧ (icache[ino].dirty ⊃ icache[ino].inode.ino ∈ files2 ∧ icache[ino].inode.nlink = files2[icache[ino].inode.ino].nlink ∧ icache[ino].inode.nsubdirs = 0 ∧ ¬ icache[ino].inode.directory ; (ino ∈ dirs1 ∨ ino ∈ files2) ∧ icache[ino].inode = (icache[ino].inode.ino ∈ dirs1 ⊃ mkinode (icache[ino].inode.ino, dirs1[icache[ino].inode.ino].meta, true, dirs1[icache[ino].inode.ino].nlink, dirs1[icache[ino].inode.ino].nsubdirs, dirs1[icache[ino].inode.ino].size) ;mkinode (icache[ino].inode.ino, files2[icache[ino].inode.ino].meta, false, files2[icache[ino].inode.ino].nlink, 0, files2[icache[ino].inode.ino].size) ) ), ∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ∈ dirs0 ∧ (links(key.ino, dirs0) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ∈ dirs0 ∨ dcache[key].ino ∈ files1) ∧ dcache[key].name ∈ dirs0[key.ino].entries ∧ dcache[key].ino = dirs0[key.ino].entries[dcache[key].name] ∧ dcache[key].dentry?, ∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ∈ dirs1 ∧ (links(key.ino, dirs1) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ∈ dirs1 ∨ dcache[key].ino ∈ files0) ∧ dcache[key].name ∈ dirs1[key.ino].entries ∧ dcache[key].ino = dirs1[key.ino].entries[dcache[key].name] ∧ dcache[key].dentry?, ∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ∈ dirs1 ∧ (links(key.ino, dirs1) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ∈ dirs1 ∨ dcache[key].ino ∈ files2) ∧ dcache[key].name ∈ dirs1[key.ino].entries ∧ dcache[key].ino = dirs1[key.ino].entries[dcache[key].name] ∧ dcache[key].dentry?, ∀ key. key ∈ pcache0 → key.data? ∧ # pcache0[key].page = VFS_PAGE_SIZE ∧ key.ino ∈ files1 ∧ (¬ pcache0[key].dirty → key.part ∈ files1[key.ino].content ∧ pcache0[key].page = files1[key.ino].content[key.part]), ∀ key. key ∈ pcache0 → key.data? ∧ # pcache0[key].page = VFS_PAGE_SIZE ∧ key.ino ∈ files0 ∧ (¬ pcache0[key].dirty → key.part ∈ files0[key.ino].content ∧ pcache0[key].page = files0[key.ino].content[key.part]) ⊦ ∃ err0. ⌊err0⌋ ∨ err0 = ESUCCESS

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 73:
⊦ pcache-cons(pcache, dirs, files) ↔ (∀ key. key ∈ pcache → key.data? ∧ # pcache[key].page = VFS_PAGE_SIZE ∧ valid-file-ino(key.ino, dirs, files) ∧ (¬ pcache[key].dirty → key.part ∈ files[key.ino].content ∧ pcache[key].page = files[key.ino].content[key.part]))
Proof step 74:
files2[m] = mkfile(m1, md1, # links(m, dirs1), content0), files0[m] = mkfile(m3, md2, # links(m, dirs1), content1), files1[m0] = mkfile(m4, md, # links(m0, dirs0), content2), afs-cons(dirs1, files2), afs-cons(dirs1, files0), afs-cons(dirs0, files1), synced(dirs1, icache, pcache), synced(dirs1, icache, pcache0), synced(dirs0, icache0, pcache0), m ∈ files0, m ∈ files2, m0 ∈ files1, ∀ key. key ∈ pcache0 → key.data? ∧ # pcache0[key].page = VFS_PAGE_SIZE ∧ key.ino ∈ files0 ∧ (¬ pcache0[key].dirty → key.part ∈ files0[key.ino].content ∧ pcache0[key].page = files0[key.ino].content[key.part]), ∀ key. key ∈ pcache0 → key.data? ∧ # pcache0[key].page = VFS_PAGE_SIZE ∧ key.ino ∈ files1 ∧ (¬ pcache0[key].dirty → key.part ∈ files1[key.ino].content ∧ pcache0[key].page = files1[key.ino].content[key.part]), ∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ∈ dirs1 ∧ (links(key.ino, dirs1) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ∈ dirs1 ∨ dcache[key].ino ∈ files2) ∧ dcache[key].name ∈ dirs1[key.ino].entries ∧ dcache[key].ino = dirs1[key.ino].entries[dcache[key].name] ∧ dcache[key].dentry?, ∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ∈ dirs1 ∧ (links(key.ino, dirs1) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ∈ dirs1 ∨ dcache[key].ino ∈ files0) ∧ dcache[key].name ∈ dirs1[key.ino].entries ∧ dcache[key].ino = dirs1[key.ino].entries[dcache[key].name] ∧ dcache[key].dentry?, ∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ∈ dirs0 ∧ (links(key.ino, dirs0) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ∈ dirs0 ∨ dcache[key].ino ∈ files1) ∧ dcache[key].name ∈ dirs0[key.ino].entries ∧ dcache[key].ino = dirs0[key.ino].entries[dcache[key].name] ∧ dcache[key].dentry?, ∀ ino. ino ∈ icache → icache[ino].inode.ino = ino ∧ (icache[ino].dirty ⊃ icache[ino].inode.ino ∈ files2 ∧ icache[ino].inode.nlink = files2[icache[ino].inode.ino].nlink ∧ icache[ino].inode.nsubdirs = 0 ∧ ¬ icache[ino].inode.directory ; (ino ∈ dirs1 ∨ ino ∈ files2) ∧ icache[ino].inode = (icache[ino].inode.ino ∈ dirs1 ⊃ mkinode (icache[ino].inode.ino, dirs1[icache[ino].inode.ino].meta, true, dirs1[icache[ino].inode.ino].nlink, dirs1[icache[ino].inode.ino].nsubdirs, dirs1[icache[ino].inode.ino].size) ;mkinode (icache[ino].inode.ino, files2[icache[ino].inode.ino].meta, false, files2[icache[ino].inode.ino].nlink, 0, files2[icache[ino].inode.ino].size) ) ), ∀ ino. ino ∈ icache → icache[ino].inode.ino = ino ∧ (icache[ino].dirty ⊃ icache[ino].inode.ino ∈ files0 ∧ icache[ino].inode.nlink = files0[icache[ino].inode.ino].nlink ∧ icache[ino].inode.nsubdirs = 0 ∧ ¬ icache[ino].inode.directory ; (ino ∈ dirs1 ∨ ino ∈ files0) ∧ icache[ino].inode = (icache[ino].inode.ino ∈ dirs1 ⊃ mkinode (icache[ino].inode.ino, dirs1[icache[ino].inode.ino].meta, true, dirs1[icache[ino].inode.ino].nlink, dirs1[icache[ino].inode.ino].nsubdirs, dirs1[icache[ino].inode.ino].size) ;mkinode (icache[ino].inode.ino, files0[icache[ino].inode.ino].meta, false, files0[icache[ino].inode.ino].nlink, 0, files0[icache[ino].inode.ino].size) ) ), ∀ ino. ino ∈ icache0 → icache0[ino].inode.ino = ino ∧ (icache0[ino].dirty ⊃ icache0[ino].inode.ino ∈ files1 ∧ icache0[ino].inode.nlink = files1[icache0[ino].inode.ino].nlink ∧ icache0[ino].inode.nsubdirs = 0 ∧ ¬ icache0[ino].inode.directory ; (ino ∈ dirs0 ∨ ino ∈ files1) ∧ icache0[ino].inode = (icache0[ino].inode.ino ∈ dirs0 ⊃ mkinode (icache0[ino].inode.ino, dirs0[icache0[ino].inode.ino].meta, true, dirs0[icache0[ino].inode.ino].nlink, dirs0[icache0[ino].inode.ino].nsubdirs, dirs0[icache0[ino].inode.ino].size) ;mkinode (icache0[ino].inode.ino, files1[icache0[ino].inode.ino].meta, false, files1[icache0[ino].inode.ino].nlink, 0, files1[icache0[ino].inode.ino].size) ) ), ∀ key. key ∈ pcache → key.data? ∧ # pcache[key].page = VFS_PAGE_SIZE ∧ valid-file-ino(key.ino, dirs1, files2) ∧ (¬ pcache[key].dirty → key.part ∈ files2[key.ino].content ∧ pcache[key].page = files2[key.ino].content[key.part]) ⊦ ∃ err0. ⌊err0⌋ ∨ err0 = ESUCCESS

Rule argument

spec: cache-base name: pcache-cons seq: ⊦ pcache-cons(pcache, dirs, files) ↔ (∀ key. key ∈ pcache → key.data? ∧ # pcache[key].page = VFS_PAGE_SIZE ∧ valid-file-ino(key.ino, dirs, files) ∧ ( ¬ pcache[key].dirty → key.part ∈ files[key.ino].content ∧ pcache[key].page = files[key.ino].content[key.part])) subst: [pcache, dirs, files] → [pcache, dirs1, files2] flags: <all paths>

Additional information

Used terms: pcache, dirs1, files2