[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 94 from proof inv-rmdir (previous step)

fs0[ri0[inodekey(m0)]].nsubdirs = m + 1, fs0[ri0[inodekey(m0)]].size = m1 + 1, 1 = fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nlink, dentrykeys(inodekey(fs0[ri0[dentrykey(m0, str)]].ino), ri0) = ∅, links(fs0[ri0[dentrykey(m0, str)]].ino, fs0, ri0) = ∅ ++ dentrykey(m0, str), replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, fs0[ri0[dentrykey(m0, str)]].dentry?, fs0[ri0[inodekey(m0)]].directory, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].directory, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].inode?, fs0[ri0[inodekey(m0)]].inode?, ro.inodes?, fo.inodes?, file-node-cons(fs0, ri0), log-cons(log0, fs0), orphans-cons(fs0, ri0, ro), replay-cons-strong(fs0, ri0, ro, fi, fo, log0), replay-safe(log0, fs0, fi ⊖ fo), ri-cons(ri0, fs0), ri-inj(ri0), ri-inj(fi), ubifs-cons(fs0, ri0, ro, fi, fo, log0), dentrykey(m0, str) ∈ ri0, inodekey(fs0[ri0[dentrykey(m0, str)]].ino) ∈ ri0, inodekey(m0) ∈ ri0, ri0[dentrykey(m0, str)] ∈ fs0, ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)] ∈ fs0, ri0[inodekey(m0)] ∈ fs0, 1 ≤ fs0[ri0[inodekey(m0)]].size, fs0[ri0[inodekey(m0)]].nlink ≠ 0 ∨ m0 = ROOT_INO ⊦ fs0[ri0[dentrykey(m0, str)]].ino = 0, fs0[ri0[dentrykey(m0, str)]].ino = ROOT_INO, fs0[ri0[inodekey(m0)]].size = 0, adr1 = adr3, adr1 = adr2, adr2 = adr3, m0 = 0, inodekey(fs0[ri0[dentrykey(m0, str)]].ino) ∈ ro, inodekey(m0) ∈ ro, adr1 ∈ fs0, adr2 ∈ fs0, adr3 ∈ fs0, replay-safe (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) ∧ ¬ adr1 ∈ replay-log (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) ∧ ¬ adr2 ∈ replay-log (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) [inodekey(m0), adr1 ] ∧ ¬ adr3 ∈ replay-log (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) [inodekey(m0), adr1 ] -- dentrykey(m0, str) ∧ replay-log (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) [inodekey(m0), adr1 ] -- dentrykey(m0, str) -- inodekey(fs0[ri0[dentrykey(m0, str)]].ino) = (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(fs0[ri0[dentrykey(m0, str)]].ino), adr3] ⊖ (ro ++ inodekey(fs0[ri0[dentrykey(m0, str)]].ino))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 95:
fs0[ri0[inodekey(m0)]].nsubdirs = m + 1, fs0[ri0[inodekey(m0)]].size = m1 + 1, 1 = fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nlink, dentrykeys(inodekey(fs0[ri0[dentrykey(m0, str)]].ino), ri0) = ∅, links(fs0[ri0[dentrykey(m0, str)]].ino, fs0, ri0) = ∅ ++ dentrykey(m0, str), replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, fs0[ri0[dentrykey(m0, str)]].dentry?, fs0[ri0[inodekey(m0)]].directory, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].directory, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].inode?, fs0[ri0[inodekey(m0)]].inode?, ro.inodes?, fo.inodes?, file-node-cons(fs0, ri0), log-cons(log0, fs0), orphans-cons(fs0, ri0, ro), replay-cons-strong(fs0, ri0, ro, fi, fo, log0), replay-safe(log0, fs0, fi ⊖ fo), ri-cons(ri0, fs0), ri-inj(ri0), ri-inj(fi), ubifs-cons(fs0, ri0, ro, fi, fo, log0), dentrykey(m0, str) ∈ ri0, inodekey(fs0[ri0[dentrykey(m0, str)]].ino) ∈ ri0, inodekey(m0) ∈ ri0, ri0[dentrykey(m0, str)] ∈ fs0, ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)] ∈ fs0, ri0[inodekey(m0)] ∈ fs0, fs0[ri0[inodekey(m0)]].nlink ≠ 0 ∨ m0 = ROOT_INO ⊦ fs0[ri0[dentrykey(m0, str)]].ino = 0, fs0[ri0[dentrykey(m0, str)]].ino = ROOT_INO, adr1 = adr3, adr1 = adr2, adr2 = adr3, m0 = 0, inodekey(fs0[ri0[dentrykey(m0, str)]].ino) ∈ ro, inodekey(m0) ∈ ro, adr1 ∈ fs0, adr2 ∈ fs0, adr3 ∈ fs0, replay-safe (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) ∧ ¬ adr1 ∈ replay-log (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) ∧ ¬ adr2 ∈ replay-log (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) [inodekey(m0), adr1 ] ∧ ¬ adr3 ∈ replay-log (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) [inodekey(m0), adr1 ] -- dentrykey(m0, str) ∧ replay-log (log0, fs0[adr1, inodenode(inodekey(m0), md0, true, fs0[ri0[inodekey(m0)]].nlink, m, m1)][adr2, dentrynode(dentrykey(m0, str), 0)] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m0, str)]].ino), md, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m0, str)]].ino)]].size) ], fi ⊖ fo) [inodekey(m0), adr1 ] -- dentrykey(m0, str) -- inodekey(fs0[ri0[dentrykey(m0, str)]].ino) = (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(fs0[ri0[dentrykey(m0, str)]].ino), adr3] ⊖ (ro ++ inodekey(fs0[ri0[dentrykey(m0, str)]].ino))

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m + n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat