[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 105 from proof ubifs-cons-reduce (previous step)

ro0.inodes?, fo0.inodes?, log-cons(log0, fs0), log-cons(log1, fs1), ri-closed(ri0, fs0, ro0), key ∈ ri0, key ∈ ri1, fo1 ⊆ dom fi1, ro1 ⊆ dom ri1, ro0 ⊆ dom ri0, fo0 ⊆ dom fi0, ∀ key0. ( key0 ∈ ri0 → (key0.inode? → fs0[ri0[key0]].inode? ∧ key0.ino ≠ 0) ∧ ( key0.dentry? → fs0[ri0[key0]].dentry? ∧ inodekey(key0.ino) ∈ ri0 ∧ inodekey(fs0[ri0[key0]].ino) ∈ ri0 ∧ ¬ inodekey(key0.ino) ∈ ro0 ∧ ¬ inodekey(fs0[ri0[key0]].ino) ∈ ro0 ∧ fs0[ri0[inodekey(key0.ino)]].directory) ∧ ( key0.data? → fs0[ri0[key0]].data? ∧ inodekey(key0.ino) ∈ ri0 ∧ ¬ fs0[ri0[inodekey(key0.ino)]].directory ∧ # fs0[ri0[key0]].data = VFS_PAGE_SIZE)) → key0 ∈ ri1 → ri1[key0] ∈ fs1 ∧ fs1[ri1[key0]].key = key0 ∧ (key0.inode? → fs1[ri1[key0]].inode? ∧ key0.ino ≠ 0) ∧ ( key0.dentry? → fs1[ri1[key0]].dentry? ∧ inodekey(key0.ino) ∈ ri1 ∧ inodekey(fs1[ri1[key0]].ino) ∈ ri1 ∧ ¬ inodekey(key0.ino) ∈ ro1 ∧ ¬ inodekey(fs1[ri1[key0]].ino) ∈ ro1 ∧ fs1[ri1[inodekey(key0.ino)]].directory) ∧ ( key0.data? → fs1[ri1[key0]].data? ∧ inodekey(key0.ino) ∈ ri1 ∧ ¬ fs1[ri1[inodekey(key0.ino)]].directory ∧ # fs1[ri1[key0]].data = VFS_PAGE_SIZE), ∀ adr. ( adr ∈ log0 → (fs0[adr].inode? → fs0[adr].key.inode? ∧ fs0[adr].key.ino ≠ 0) ∧ (fs0[adr].dentry? → fs0[adr].key.dentry?) ∧ (fs0[adr].data? → fs0[adr].key.data?) ∧ (fs0[adr].trunc? → fs0[adr].key.inode?)) → adr ∈ log1 → adr ∈ fs1 ∧ (fs1[adr].inode? → fs1[adr].key.inode? ∧ fs1[adr].key.ino ≠ 0) ∧ (fs1[adr].dentry? → fs1[adr].key.dentry?) ∧ (fs1[adr].data? → fs1[adr].key.data?) ∧ (fs1[adr].trunc? → fs1[adr].key.inode?), ∀ key. key ∈ fo1 → key ∈ fi1, ∀ key. key ∈ ro1 → key ∈ ri1, ∀ key. key ∈ ri0 → ri0[key] ∈ fs0 ∧ key-cons(key, fs0[ri0[key]]) ⊦ dups(log0), inodekey(0) ∈ ri1, inodekey(0) ∈ ri0, (key.inode? → fs0[ri0[key]].inode? ∧ key.ino ≠ 0) ∧ ( key.dentry? → fs0[ri0[key]].dentry? ∧ inodekey(key.ino) ∈ ri0 ∧ inodekey(fs0[ri0[key]].ino) ∈ ri0 ∧ ¬ inodekey(key.ino) ∈ ro0 ∧ ¬ inodekey(fs0[ri0[key]].ino) ∈ ro0 ∧ fs0[ri0[inodekey(key.ino)]].directory) ∧ (key.data? → fs0[ri0[key]].data? ∧ inodekey(key.ino) ∈ ri0 ∧ ¬ fs0[ri0[inodekey(key.ino)]].directory ∧ # fs0[ri0[key]].data = VFS_PAGE_SIZE), ri1[key] ∈ fs1 ∧ key-cons(key, fs1[ri1[key]])

User applied rule all left

And obtained the following new goals

Proof step 106:
fo0.inodes?, ro0.inodes?, log-cons(log1, fs1), log-cons(log0, fs0), ri-closed(ri0, fs0, ro0), key ∈ ri1, key ∈ ri0, fo0 ⊆ dom fi0, ro0 ⊆ dom ri0, ro1 ⊆ dom ri1, fo1 ⊆ dom fi1, key ∈ ri0 → ri0[key] ∈ fs0 ∧ key-cons(key, fs0[ri0[key]]), ∀ key. key ∈ ro1 → key ∈ ri1, ∀ key. key ∈ fo1 → key ∈ fi1, ∀ adr. ( adr ∈ log0 → (fs0[adr].inode? → fs0[adr].key.inode? ∧ fs0[adr].key.ino ≠ 0) ∧ (fs0[adr].dentry? → fs0[adr].key.dentry?) ∧ (fs0[adr].data? → fs0[adr].key.data?) ∧ (fs0[adr].trunc? → fs0[adr].key.inode?)) → adr ∈ log1 → adr ∈ fs1 ∧ (fs1[adr].inode? → fs1[adr].key.inode? ∧ fs1[adr].key.ino ≠ 0) ∧ (fs1[adr].dentry? → fs1[adr].key.dentry?) ∧ (fs1[adr].data? → fs1[adr].key.data?) ∧ (fs1[adr].trunc? → fs1[adr].key.inode?), ∀ key0. ( key0 ∈ ri0 → (key0.inode? → fs0[ri0[key0]].inode? ∧ key0.ino ≠ 0) ∧ ( key0.dentry? → fs0[ri0[key0]].dentry? ∧ inodekey(key0.ino) ∈ ri0 ∧ inodekey(fs0[ri0[key0]].ino) ∈ ri0 ∧ ¬ inodekey(key0.ino) ∈ ro0 ∧ ¬ inodekey(fs0[ri0[key0]].ino) ∈ ro0 ∧ fs0[ri0[inodekey(key0.ino)]].directory) ∧ ( key0.data? → fs0[ri0[key0]].data? ∧ inodekey(key0.ino) ∈ ri0 ∧ ¬ fs0[ri0[inodekey(key0.ino)]].directory ∧ # fs0[ri0[key0]].data = VFS_PAGE_SIZE)) → key0 ∈ ri1 → ri1[key0] ∈ fs1 ∧ fs1[ri1[key0]].key = key0 ∧ (key0.inode? → fs1[ri1[key0]].inode? ∧ key0.ino ≠ 0) ∧ ( key0.dentry? → fs1[ri1[key0]].dentry? ∧ inodekey(key0.ino) ∈ ri1 ∧ inodekey(fs1[ri1[key0]].ino) ∈ ri1 ∧ ¬ inodekey(key0.ino) ∈ ro1 ∧ ¬ inodekey(fs1[ri1[key0]].ino) ∈ ro1 ∧ fs1[ri1[inodekey(key0.ino)]].directory) ∧ ( key0.data? → fs1[ri1[key0]].data? ∧ inodekey(key0.ino) ∈ ri1 ∧ ¬ fs1[ri1[inodekey(key0.ino)]].directory ∧ # fs1[ri1[key0]].data = VFS_PAGE_SIZE) ⊦ dups(log0), inodekey(0) ∈ ri0, inodekey(0) ∈ ri1, ri1[key] ∈ fs1 ∧ key-cons(key, fs1[ri1[key]]), (key.inode? → fs0[ri0[key]].inode? ∧ key.ino ≠ 0) ∧ ( key.dentry? → fs0[ri0[key]].dentry? ∧ inodekey(key.ino) ∈ ri0 ∧ inodekey(fs0[ri0[key]].ino) ∈ ri0 ∧ ¬ inodekey(key.ino) ∈ ro0 ∧ ¬ inodekey(fs0[ri0[key]].ino) ∈ ro0 ∧ fs0[ri0[inodekey(key.ino)]].directory) ∧ (key.data? → fs0[ri0[key]].data? ∧ inodekey(key.ino) ∈ ri0 ∧ ¬ fs0[ri0[inodekey(key.ino)]].directory ∧ # fs0[ri0[key]].data = VFS_PAGE_SIZE)

Rule argument

pos: left 16 terms: key abort: F discard: T computed: T

Additional information

Used terms: key