This is a listing of all goals defined over the current specification. Axioms are marked yellow, proved theorems green and unproved theorems blue. Click on individual names to get more information on the proof.
| Name: | Sequent: | Type: |
|---|---|---|
| datakeys-set | ⊦ key ∈ datakeys(ks, ri) ↔ key ∈ ri ∧ key.data? ∧ inodekey(key.ino) ∈ ks | An Axiom |
| datakeys-ge | ⊦ key ∈ datakeys(key0, n, ri) ↔ key ∈ datakeys(key0, ri) ∧ n ≤ key.part * VFS_PAGE_SIZE | An Axiom |
| datakeys | ⊦ key ∈ datakeys(key0, ri) ↔ key ∈ ri ∧ key.data? ∧ key0.inode? ∧ key.ino = key0.ino | An Axiom |
| dentrykeys | ⊦ key ∈ dentrykeys(key0, ri) ↔ key ∈ ri ∧ key.dentry? ∧ key0.inode? ∧ key.ino = key0.ino | An Axiom |
| dentries | ⊦ str ∈ dentries(key, ri) ↔ dentrykey(key.ino, str) ∈ dentrykeys(key, ri) | An Axiom |
| dels-with-data | ⊦ ri ⊖ ks = (ri \ ks) \ datakeys(ks, ri) | An Axiom |
| del-with-data | ⊦ ri ⊖ key = (ri -- key) \ datakeys(key, ri) | An Axiom |
| valid-negdentry-with-parent | ⊦ valid-negdentry(ino, dent, fs, ri, ro) ↔ dent.negdentry? ∧ valid-parent-ino(ino, fs, ri, ro) ∧ ¬ dentrykey(ino, dent.name) ∈ ri | An Axiom |
| valid-dentry-with-parent | ⊦ valid-dentry(ino, dent, fs, ri, ro) ↔ dent.dentry? ∧ valid-parent-ino(ino, fs, ri, ro) ∧ valid-ino(dent.ino, fs, ri) ∧ dentrykey(ino, dent.name) ∈ ri ∧ fs[ri[dentrykey(ino, dent.name)]].ino = dent.ino | An Axiom |
| valid-dentry | ⊦ valid-dentry(dent, fs, ri, ro) ↔ (∃ ino. valid-dentry(ino, dent, fs, ri, ro)) | An Axiom |
| valid-parent-ino | ⊦ valid-parent-ino(ino, fs, ri, ro) ↔ (valid-dir-ino(ino, fs, ri) ∧ (fs[ri[inodekey(ino)]].nlink ≠ 0 ∨ ino = ROOT_INO)) ∧ ¬ inodekey(ino) ∈ ro | An Axiom |
| valid-dir-ino | ⊦ valid-dir-ino(ino, fs, ri) ↔ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory | An Axiom |
| valid-file-ino | ⊦ valid-file-ino(ino, fs, ri) ↔ inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory | An Axiom |
| valid-ino | ⊦ valid-ino(ino, fs, ri) ↔ inodekey(ino) ∈ ri | An Axiom |
| links | ⊦ key ∈ links(ino, fs, ri) ↔ key ∈ ri ∧ key.dentry? ∧ fs[ri[key]].ino = ino | An Axiom |
| new-rec | ⊦ new(adr ' + log, fs) ↔ ¬ adr ∈ fs ∧ new(log, fs) | An Axiom |
| new-base | ⊦ new([], fs) | An Axiom |
| index-update-key-rec | ⊦ ri[key ' + kx, adr ' + ax] = ri[key, adr][kx, ax] | An Axiom |
| index-update-key-base | ⊦ ri[[], []] = ri | An Axiom |
| keys-rec | ⊦ keys(fs, adr ' + ax) = fs[adr].key + keys(fs, ax) | An Axiom |
| keys-base | ⊦ keys(fs, []) = [] | An Axiom |
| range | ⊦ adr ∈ range ri ↔ adr ∈ ri | An Axiom |
| aubifs-crash | ⊦ aubifs-crash(ri0, ld0, ro0, fi0, fo0, fs0, log0, ri1, ld1, ro1, fi1, fo1, fs1, log1) ↔ fi1 = fi0 ∧ fo1 = fo0 ∧ fs1 = fs0 ∧ log1 = log0 | An Axiom |
| index-lookup-rec | ⊦ ri[key ' + kx] = ri[key] + ri[kx] | An Axiom |
| index-lookup-base | ⊦ ri[[]] = [] | An Axiom |
| read-rec | ⊦ fs[adr ' + ax] = fs[adr] + fs[ax] | An Axiom |
| read-base | ⊦ fs[[]] = [] | An Axiom |
| write-rec | ⊦ fs[adr ' + ax, nd ' + ndx] = fs[adr, nd][ax, ndx] | An Axiom |
| write-base | ⊦ fs[[], []] = fs | An Axiom |
| at-del-ro-data | ⊦ datakey(ino, n) ∈ ri ∧ datakey(ino, n) ≠ key0 ∧ inodekey(ino) ≠ key0 → (ri ⊖ key0)[datakey(ino, n)] = ri[datakey(ino, n)] | A User Theorem |
| at-del-ro-nodata | ⊦ key ∈ ri ∧ key ≠ key0 ∧ ¬ key.data? → (ri ⊖ key0)[key] = ri[key] | A User Theorem |
| at-dels-ro-data | ⊦ datakey(ino, n) ∈ ri ∧ ¬ datakey(ino, n) ∈ ks ∧ ¬ inodekey(ino) ∈ ks → (ri ⊖ ks)[datakey(ino, n)] = ri[datakey(ino, n)] | A User Theorem |
| at-dels-ro-nodata | ⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key] | A User Theorem |
| at-write-at | ⊦ # ax = # ndx ∧ adr ∈ ax ∧ ¬ dups(ax) → fs[ax, ndx][adr] = ndx[pos(adr, ax)] | A User Theorem |
| at-write-other | ⊦ # ax = # ndx ∧ ¬ adr ∈ ax → fs[ax, ndx][adr] = fs[adr] | A User Theorem |
| data-at-del-datakeys | ⊦ ino0 ≠ ino → (ri \ datakeys(inodekey(ino0), ri))[datakey(ino, n)] = ri[datakey(ino, n)] | A User Theorem |
| data-in-empty | ⊦ ¬ adr ∈ ∅ | A User Theorem |
| datakeys-0 | ⊦ datakeys(key, 0, ri) = datakeys(key, ri) | A User Theorem |
| datakeys-del-dentry | ⊦ datakeys(key, ri -- dentrykey(ino, str)) = datakeys(key, ri) | A User Theorem |
| datakeys-put-dentry | ⊦ datakeys(ks, ri[dentrykey(n, str), adr]) = datakeys(ks, ri) | A User Theorem |
| datakeys-put-inode | ⊦ datakeys(ks, ri[inodekey(n), adr]) = datakeys(ks, ri) | A User Theorem |
| datakeys-put-inode-01 | ⊦ datakeys(key, ri[inodekey(ino), adr]) = datakeys(key, ri) | A User Theorem |
| datakeys-set-add | ⊦ datakeys(ks ++ key, ri) = datakeys(ks, ri) ∪ datakeys(key, ri) | A User Theorem |
| datakeys-set-del | ⊦ datakeys(ks -- key, ri) = datakeys(ks, ri) \ datakeys(key, ri) | A User Theorem |
| datakeys-set-empty | ⊦ datakeys(∅, ri) = ∅ | A User Theorem |
| del-commute-dels | ⊦ ¬ key ∈ ks → (ri ⊖ key) ⊖ ks = (ri ⊖ ks) ⊖ key | A User Theorem |
| del-del-again | ⊦ inodekey(m) ∈ ks → (ri ⊖ ks) ⊖ inodekey(m) = ri ⊖ ks | A User Theorem |
| del-merge-dels | ⊦ key ∈ ks → (ri ⊖ key) ⊖ ks = ri ⊖ ks | A User Theorem |
| del-put-again | ⊦ inodekey(m) ∈ ks → ri[inodekey(m), adr] ⊖ ks = ri ⊖ ks | A User Theorem |
| del-this | ⊦ ri[key, adr] ⊖ key = ri ⊖ key | A User Theorem |
| del-with-emp | ⊦ ri ⊖ ∅ = ri | A User Theorem |
| dels-ignore-data | ⊦ inodekey(ino) ∈ ro → (ri -- datakey(ino, n)) ⊖ ro = ri ⊖ ro | A User Theorem |
| dentry-at-del-datakeys | ⊦ (ri \ datakeys(key, ri))[dentrykey(ino, str)] = ri[dentrykey(ino, str)] | A User Theorem |
| in-del-ro | ⊦ key ∈ ri ⊖ key0 ↔ key ∈ ri ∧ key ≠ key0 ∧ (key.data? → inodekey(key.ino) ≠ key0) | A User Theorem |
| in-dels-inode | ⊦ inodekey(ino) ∈ ri → (inodekey(ino) ∈ ri ⊖ ro ↔ ¬ inodekey(ino) ∈ ro) | A User Theorem |
| in-dels-ro | ⊦ key ∈ ri ⊖ ks ↔ key ∈ ri ∧ ¬ key ∈ ks ∧ (key.data? → ¬ inodekey(key.ino) ∈ ks) | A User Theorem |
| in-read-in-dom | ⊦ kx.toSet ⊆ dom ri ∧ adr ∈ ri[kx] → adr ∈ ri | A User Theorem |
| in-write | # ax = # ndx ⊦ adr ∈ fs[ax, ndx] ↔ adr ∈ fs ∨ adr ∈ ax | A User Theorem |
| index-lookup-app | ⊦ ri[kx0 + kx1] = ri[kx0] + ri[kx1] | A User Theorem |
| index-lookup-length | ⊦ # ri[kx] = # kx | A User Theorem |
| index-lookup-one | ⊦ ri[key '] = ri[key] ' | A User Theorem |
| index-update-at-last-oc | ⊦ # ax = # kx0 + # kx1 + 1 ∧ ¬ key ∈ kx1 → ri[kx0 + key ' + kx1, ax][key] = ax[# kx0] | A User Theorem |
| index-update-dom | # kx = # ax ⊦ dom ri[kx, ax] = dom ri ∪ kx.toSet | A User Theorem |
| index-update-in | ⊦ # kx = # ax → (key ∈ ri[kx, ax] ↔ key ∈ ri ∨ key ∈ kx) | A User Theorem |
| inode-at-del-datakeys | ⊦ (ri \ datakeys(key, ri))[inodekey(ino)] = ri[inodekey(ino)] | A User Theorem |
| keys-append | ⊦ keys(fs, ax0 + ax1) = keys(fs, ax0) + keys(fs, ax1) | A User Theorem |
| keys-at-other | ⊦ # kx = # ax ∧ key ∈ ri ∧ ¬ key ∈ kx → ri[kx, ax][key] = ri[key] | A User Theorem |
| keys-drop-fs | ⊦ ¬ adr ∈ ax → keys(fs[adr, nd], ax) = keys(fs, ax) | A User Theorem |
| keys-firstn | ⊦ n ≤ # ax → firstn(n, keys(fs, ax)) = keys(fs, firstn(n, ax)) | A User Theorem |
| keys-in-elim | ⊦ key ∈ keys(fs, ax) ↔ (∃ adr. adr ∈ ax ∧ fs[adr].key = key) | A User Theorem |
| keys-in-elim-last-oc | ⊦ key ∈ keys(fs, ax) ↔ (∃ ax0, ax1, adr. ax = ax0 + adr ' + ax1 ∧ fs[adr].key = key ∧ ¬ key ∈ keys(fs, ax1)) | A User Theorem |
| keys-length | ⊦ # keys(fs, ax) = # ax | A User Theorem |
| keys-one | ⊦ keys(fs, adr ') = fs[adr].key ' | A User Theorem |
| new-append | ⊦ new(ax0 + ax1, fs) ↔ new(ax0, fs) ∧ new(ax1, fs) | A User Theorem |
| new-butlastn | ⊦ n ≤ # ax ∧ new(ax, fs) → new(butlastn(n, ax), fs) | A User Theorem |
| new-ext | ⊦ new(log, fs) ↔ (∀ adr. adr ∈ log → ¬ adr ∈ fs) | A User Theorem |
| new-in-fs | ⊦ new(log, fs) ∧ adr ∈ fs → ¬ adr ∈ log | A User Theorem |
| new-in-log | ⊦ new(log, fs) ∧ adr ∈ log → ¬ adr ∈ fs | A User Theorem |
| new-inode-exists | ⊦ ∃ ino. ¬ inodekey(ino) ∈ ks | A User Theorem |
| new-not-new | ⊦ new(ax, fs) ∧ adr ∈ ax → (adr ∈ fs ↔ false) | A User Theorem |
| new-one | ⊦ new(adr ', fs) ↔ ¬ adr ∈ fs | A User Theorem |
| new-write | ⊦ new(ax, fs) → (new(ax, fs[adr, nd]) ↔ ¬ adr ∈ ax) | A User Theorem |
| nodups-new | # ax0 = # ndx0 ⊦ ¬ dups(ax0) ∧ ¬ dups(ax1) ∧ new(ax1, fs1) ∧ fs1 = fs0[ax0, ndx0] → ¬ dups(ax0 + ax1) | A User Theorem |
| notin-del | ⊦ ¬ adr ∈ ri → ¬ adr ∈ ri ⊖ key | A User Theorem |
| notin-dels | ⊦ ¬ adr ∈ ri → ¬ adr ∈ ri ⊖ ks | A User Theorem |
| range-empty | ⊦ range ∅ = ∅ | A User Theorem |
| range-inter-empty-in | ⊦ key ∈ ri ∧ as ∩ range ri = ∅ → ¬ ri[key] ∈ as | A User Theorem |
| read-app | ⊦ fs[ax0 + ax1] = fs[ax0] + fs[ax1] | A User Theorem |
| read-eq-empty | ⊦ fs[ax] = [] ↔ ax = [] | A User Theorem |
| read-eq-one | ⊦ fs[ax] = nd ' ↔ (∃ adr. ax = adr ' ∧ fs[adr] = nd) | A User Theorem |
| read-length | ⊦ # fs[ax] = # ax | A User Theorem |
| read-one | ⊦ fs[adr '] = fs[adr] ' | A User Theorem |
| read-split | ⊦ fs[ax] = ndx0 + ndx1 ↔ (∃ ax0, ax1. ax = ax0 + ax1 ∧ fs[ax0] = ndx0 ∧ fs[ax1] = ndx1) | A User Theorem |
| read-write-other | ⊦ ¬ key ∈ kx → ri[key, adr][kx] = ri[kx] | A User Theorem |
| read-write-other-fs | ⊦ ¬ adr ∈ ax → fs[adr, nd][ax] = fs[ax] | A User Theorem |
| toSet-subseteq-domri-notin | ⊦ kx.toSet ⊆ dom ri ∧ ¬ key ∈ ri → ¬ key ∈ kx | A User Theorem |
| update-index-app | # kx = # ax ⊦ ri[kx + key ', ax + adr '] = ri[kx, ax][key, adr] | A User Theorem |
| update-index-append | # kx0 = # ax0, # kx1 = # ax1 ⊦ ri[kx0 + kx1, ax0 + ax1] = ri[kx0, ax0][kx1, ax1] | A User Theorem |
| update-index-data-notin | # kx = # ax ⊦ ¬ adr ∈ ax ∧ ¬ adr ∈ ri → ¬ adr ∈ ri[kx, ax] | A User Theorem |
| update-index-one | ⊦ ri[key ', adr '] = ri[key, adr] | A User Theorem |
| update-index-shift | # ax = # kx ⊦ ¬ key ∈ kx → ri[key, adr][kx, ax] = ri[kx, ax][key, adr] | A User Theorem |
| write-app | # ax0 = # ndx0, # ax1 = # ndx1 ⊦ fs[ax0 + ax1, ndx0 + ndx1] = fs[ax0, ndx0][ax1, ndx1] | A User Theorem |
| write-dom | # ax = # ndx ⊦ dom fs[ax, ndx] = dom fs ∪ ax.toSet | A User Theorem |
| write-new | # ax0 = # ndx ⊦ new(ax1, fs1) ∧ fs1 = fs0[ax0, ndx] → new(ax1, fs0) | A User Theorem |
| write-new-app | # ax0 = # ndx0 ⊦ new(ax0, fs0) ∧ new(ax1, fs0[ax0, ndx0]) → new(ax0 + ax1, fs0) | A User Theorem |
| write-new-app-nodups | # ax0 = # ndx0 ⊦ ¬ dups(ax0) ∧ ¬ dups(ax1) ∧ new(ax1, fs0[ax0, ndx0]) → ¬ dups(ax0 + ax1) | A User Theorem |
| write-one | ⊦ fs[adr ', nd '] = fs[adr, nd] | A User Theorem |
| write-shift | # ax = # ndx ⊦ ¬ adr ∈ ax → fs[adr, nd][ax, ndx] = fs[ax, ndx][adr, nd] | A User Theorem |