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Proof step 148 from proof refine-rmdir (previous step)

fs0[ri0[inodekey(m2)]] = inodenode(key1, md2, true, 1, m5, m6), fs0[ri0[dentrykey(m, str)]] = dentrynode(key, m2), fs0[ri0[inodekey(m)]] = inodenode(key0, md1, true, m3, m0 + 1, m1 + 1), abs-children(m2, fs0, ri0) = ∅, dentrykeys(inodekey(m2), ri0) = ∅, dirs = abs-dirs(fs0, ri0), files = abs-files(fs0, ri0), links(m2, fs0, ri0) = ∅ ++ dentrykey(m, str), old_dent = mkdentry(str, m2), p_inode0 = mkinode(m, md, true, m3, m0 + 1, m1 + 1), p_inode1 = mkinode(m2, md0, true, 1, m5, m6), orphans(fs0, ri0).inodes?, fo.inodes?, afs-cons(abs-dirs(fs0, ri0), abs-files(fs0, ri0)), afs-cons(dirs, files), file-node-cons(fs0, ri0), file-node-cons (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)][adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6)], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]), log-cons(log0, fs0), log-cons (log0, fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)][adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6)]), orphans-cons(fs0, ri0, orphans(fs0, ri0)), orphans-cons (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)][adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6)], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3], orphans(fs0, ri0) ++ inodekey(m2)), pre-rmdir(p_inode0, p_inode1, old_dent, dirs, files), replay-cons-strong(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), replay-cons-strong (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)][adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6)], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3], orphans(fs0, ri0) ++ inodekey(m2), fi, fo, ((log0 + adr1 ') + adr2 ') + adr3 '), ri-cons(ri0, fs0), ri-inj(ri0), ri-inj(fi), ri-inj((ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]), ubifs-cons(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), ubifs-cons (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)][adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6)], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3], orphans(fs0, ri0) ++ inodekey(m2), fi, fo, ((log0 + adr1 ') + adr2 ') + adr3 '), inodekey(m) ∈ ri0, ri0[inodekey(m)] ∈ fs0, ri0[inodekey(m2)] ∈ fs0, inodekey(m2) ∈ ri0, ri0[dentrykey(m, str)] ∈ fs0, dentrykey(m, str) ∈ ri0, asynced → boolvar0, links(m, abs-dirs(fs0, ri0)) ≠ ∅ ∨ m = ROOT_INO, m3 ≠ 0 ∨ m = ROOT_INO ⊦ ⟨afs_rmdir#(; p_inode0, p_inode1, old_dent, err0, dirs, files; asynced)⟩ ( afs-cons(dirs, files) → dirs = abs-dirs (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]) ∧ files = abs-files (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]) ∧ orphans (abs-dirs (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]), abs-files (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, m5, m6) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3])) = abs-keys(orphans(fs0, ri0)) ++ m2 ∧ (asynced → boolvar) ∧ mkinode(m, md, true, m3, m0, m1) = p_inode0 ∧ mkinode(m2, md0, true, 0, m5, m6) = p_inode1 ∧ negdentry(str) = old_dent ∧ ESUCCESS = err0), adr1 = adr3, adr1 = adr2, adr2 = adr3, m = 0, m2 = 0, m2 = ROOT_INO, adr1 ∈ fs0, adr2 ∈ fs0, adr3 ∈ fs0, m3 = 0 ∧ m ≠ ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 149:
1 = fs0[ri0[inodekey(m2)]].nlink, fs0[ri0[inodekey(m2)]] = inodenode(key1, md2, true, 1, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size), fs0[ri0[dentrykey(m, str)]] = dentrynode(key, m2), fs0[ri0[inodekey(m)]] = inodenode(key0, md1, true, m3, m0 + 1, m1 + 1), abs-children(m2, fs0, ri0) = ∅, dentrykeys(inodekey(m2), ri0) = ∅, dirs = abs-dirs(fs0, ri0), files = abs-files(fs0, ri0), links(m2, fs0, ri0) = ∅ ++ dentrykey(m, str), old_dent = mkdentry(str, m2), p_inode0 = mkinode(m, md, true, m3, m0 + 1, m1 + 1), p_inode1 = mkinode(m2, md0, true, 1, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size), adr1 ≠ adr3, adr1 ≠ adr2, adr2 ≠ adr3, m ≠ 0, m2 ≠ 0, m2 ≠ ROOT_INO, fs0[ri0[inodekey(m2)]].directory, orphans(fs0, ri0).inodes?, fo.inodes?, afs-cons(abs-dirs(fs0, ri0), abs-files(fs0, ri0)), file-node-cons(fs0, ri0), file-node-cons (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]), log-cons(log0, fs0), log-cons (log0, fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) ]), orphans-cons(fs0, ri0, orphans(fs0, ri0)), orphans-cons (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3], orphans(fs0, ri0) ++ inodekey(m2)), replay-cons-strong(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), replay-cons-strong (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3], orphans(fs0, ri0) ++ inodekey(m2), fi, fo, ((log0 + adr1 ') + adr2 ') + adr3 '), ri-cons(ri0, fs0), ri-inj(ri0), ri-inj(fi), ri-inj((ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]), ubifs-cons(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), ubifs-cons (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3], orphans(fs0, ri0) ++ inodekey(m2), fi, fo, ((log0 + adr1 ') + adr2 ') + adr3 '), inodekey(m) ∈ ri0, ri0[inodekey(m)] ∈ fs0, ri0[inodekey(m2)] ∈ fs0, inodekey(m2) ∈ ri0, ri0[dentrykey(m, str)] ∈ fs0, dentrykey(m, str) ∈ ri0, ¬ adr1 ∈ fs0, ¬ adr2 ∈ fs0, ¬ adr3 ∈ fs0, asynced → boolvar0, links(m, abs-dirs(fs0, ri0)) ≠ ∅ ∨ m = ROOT_INO, m3 ≠ 0 ∨ m = ROOT_INO ⊦ ⟨afs_rmdir#(; p_inode0, p_inode1, old_dent, err0, dirs, files; asynced)⟩ ( afs-cons(dirs, files) → dirs = abs-dirs (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]) ∧ files = abs-files (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]) ∧ orphans (fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m2), md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]) = orphans(fs0, ri0) ++ inodekey(m2) ∧ (asynced → boolvar) ∧ mkinode(m, md, true, m3, m0, m1) = p_inode0 ∧ mkinode(m2, md0, true, 0, fs0[ri0[inodekey(m2)]].nsubdirs, fs0[ri0[inodekey(m2)]].size) = p_inode1 ∧ negdentry(str) = old_dent ∧ ESUCCESS = err0), m3 = 0 ∧ m ≠ ROOT_INO

Rule argument

Additional information

Used simplifier rules: ⊦ inodenode(key, md, boolvar, m, m0, m1).directory ↔ boolvar; (s)% ⊦ inodenode(key, md, boolvar, m, m0, m1).nlink = m; (s)% ⊦ inodenode(key, md, boolvar, m, m0, m1).size = m1; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ inodenode(key, md, boolvar, m, m0, m1).nsubdirs = m0; (s)% ⊦ dentrynode(key, m).ino = m; (s)%⊦ abs-keys(orphans(fs0, ri0)) = abs-keys(orphans(fs, ri)) ++ n ↔ orphans(fs0, ri0) = orphans(fs, ri) ++ inodekey(n); (s)%⊦ afs-cons(abs-dirs(fs, ri), abs-files(fs, ri)) → orphans(abs-dirs(fs, ri), abs-files(fs, ri)) = abs-keys(orphans(fs, ri)); (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).directory ↔ boolvar; (s)% ⊦ mkdir(m, md, m0, m1, entries).nsubdirs = m1; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nsubdirs = m1; (s)% ⊦ mkdir(m, md, m0, m1, entries).size = m; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).size = m2; (s)% ⊦ mkdir(m, md, m0, m1, entries).nlink = m0; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nlink = m0; (s)% ⊦ valid-dir-ino(ino, dirs, files) ↔ ino ∈ dirs; (s)%⊦ valid-dir-inode(inode, dirs, files) ↔ valid-dir-ino(inode.ino, dirs, files) ∧ inode.nlink = dirs[inode.ino].nlink ∧ inode.size = dirs[inode.ino].size ∧ inode.nsubdirs = dirs[inode.ino].nsubdirs ∧ inode.directory; (s)% ⊦ dir-empty(ino, dirs, files) ↔ dirs[ino].entries = ∅; (s)% ⊦ mkdentry(str, m).dentry?; (s)%⊦ dentrykey(ino, str) ∈ ri → abs-children(ino, fs, ri)[str] = fs[ri[dentrykey(ino, str)]].ino; (s)% ⊦ str ∈ abs-children(ino, fs, ri) ↔ dentrykey(ino, str) ∈ ri; (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)%⊦ abs-dir(ino, fs, ri) = mkdir (fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, fs[ri[inodekey(ino)]].nsubdirs, abs-children(ino, fs, ri)); (s)%⊦ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → abs-dirs(fs, ri)[ino] = abs-dir(ino, fs, ri); (s)% ⊦ mkdentry(str, m).name = str; (s)%⊦ ino ∈ abs-files(fs, ri) ↔ inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)%⊦ ino ∈ abs-dirs(fs, ri) ↔ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)%⊦ valid-dentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ valid-ino(dent.ino, dirs, files) ∧ dent.name ∈ dirs[ino].entries ∧ dent.ino = dirs[ino].entries[dent.name] ∧ dent.dentry?; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)% ⊦ mkdentry(str, m).ino = m; (s)%⊦ pre-rmdir(p_inode, c_inode, dent, dirs, files) ↔ dent.ino ≠ ROOT_INO ∧ valid-dentry(p_inode.ino, dent, dirs, files) ∧ dir-empty(dent.ino, dirs, files) ∧ valid-dir-inode(p_inode, dirs, files) ∧ valid-dir-inode(c_inode, dirs, files) ∧ c_inode.ino = dent.ino; (s)