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Proof step 112 from proof refine-rmdir (previous step)

1 = fs0[ri0[inodekey(fs0[ri0[dentrykey(m, str)]].ino)]].nlink, abs-children(fs0[ri0[dentrykey(m, str)]].ino, fs0, ri0) = ∅, adr = adr1, dent = mkdentry(str, fs0[ri0[dentrykey(m, str)]].ino), dentrykeys(inodekey(fs0[ri0[dentrykey(m, str)]].ino), ri0) = ∅, dirs = abs-dirs(fs0, ri0), files = abs-files(fs0, ri0), fs = fs0[adr1, inodenode(inodekey(m), md, true, fs0[ri0[inodekey(m)]].nlink, fs0[ri0[inodekey(m)]].nsubdirs - 1, fs0[ri0[inodekey(m)]].size - 1)] [adr2, dentrynode(dentrykey(m, str), 0) ] [adr3, inodenode (inodekey(fs0[ri0[dentrykey(m, str)]].ino), md0, true, 0, fs0[ri0[inodekey(fs0[ri0[dentrykey(m, str)]].ino)]].nsubdirs, fs0[ri0[inodekey(fs0[ri0[dentrykey(m, str)]].ino)]].size) ], key = inodekey(m), links(fs0[ri0[dentrykey(m, str)]].ino, fs0, ri0) = ∅ ++ dentrykey(m, str), log = ((log0 + adr1 ') + adr2 ') + adr3 ', m0 = fs0[ri0[inodekey(m)]].nlink, m1 = fs0[ri0[inodekey(m)]].nsubdirs, m2 = fs0[ri0[inodekey(m)]].size, m3 = fs0[ri0[dentrykey(m, str)]].ino, m5 = fs0[ri0[inodekey(fs0[ri0[dentrykey(m, str)]].ino)]].nsubdirs, m6 = fs0[ri0[inodekey(fs0[ri0[dentrykey(m, str)]].ino)]].size, ri = ri0[key, adr], ro = orphans(fs0, ri0), ro0 = orphans(fs0, ri0), fs0[ri0[inodekey(m)]].directory, fs0[ri0[inodekey(fs0[ri0[dentrykey(m, str)]].ino)]].directory, fo.inodes?, orphans(fs0, ri0).inodes?, afs-cons(abs-dirs(fs0, ri0), abs-files(fs0, ri0)), file-node-cons(fs0, ri0), log-cons(log0, fs0), orphans-cons(fs0, ri0, orphans(fs0, ri0)), replay-cons-strong(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), ri-cons(ri0, fs0), ri-inj(fi), ri-inj(ri0), ubifs-cons(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), dentrykey(m, str) ∈ ri0, inodekey(fs0[ri0[dentrykey(m, str)]].ino) ∈ ri0, inodekey(m) ∈ ri0, 1 ≤ fs0[ri0[inodekey(m)]].size, 1 ≤ fs0[ri0[inodekey(m)]].nsubdirs, asynced → boolvar0, fs0[ri0[inodekey(m)]].nlink ≠ 0 ∨ m = ROOT_INO, links(m, abs-dirs(fs0, ri0)) ≠ ∅ ∨ m = ROOT_INO ⊦ [aubifs_core_index_remove#(dentrykey(m, dent.name), fi, ro0, fo, fs, ((log0 + adr1 ') + adr2 ') + adr3 ', boolvar; ri); aubifs_core_index_store#(inodekey(m3), adr3, fi, ro0, fo, fs, ((log0 + adr1 ') + adr2 ') + adr3 ', boolvar; ri); aubifs_core_orphans_insert#(inodekey(m3), ri, fi, fo, fs, ((log0 + adr1 ') + adr2 ') + adr3 ', boolvar; ro0); dent := negdentry(dent.name); p_inode := mkinode (m, inodenode(inodekey(m), md, true, m0, m1 - 1, m2 - 1).meta, inodenode(inodekey(m), md, true, m0, m1 - 1, m2 - 1).directory, inodenode(inodekey(m), md, true, m0, m1 - 1, m2 - 1).nlink, inodenode(inodekey(m), md, true, m0, m1 - 1, m2 - 1).nsubdirs, inodenode(inodekey(m), md, true, m0, m1 - 1, m2 - 1).size); c_inode := mkinode (m3, inodenode(inodekey(m3), md0, true, 0, m5, m6).meta, inodenode(inodekey(m3), md0, true, 0, m5, m6).directory, inodenode(inodekey(m3), md0, true, 0, m5, m6).nlink, inodenode(inodekey(m3), md0, true, 0, m5, m6).nsubdirs, inodenode(inodekey(m3), md0, true, 0, m5, m6).size)] ( ubifs-cons(fs, ri, ro0, fi, fo, ((log0 + adr1 ') + adr2 ') + adr3 ') ∧ replay-cons-strong(fs, ri, ro0, fi, fo, ((log0 + adr1 ') + adr2 ') + adr3 ') ∧ file-node-cons(fs, ri) ∧ orphans-cons(fs, ri, ro0) ∧ ri-inj(ri) ∧ log-cons(log0, fs) ∧ adr1 ∈ fs ∧ node-cons(fs[adr1].key, fs[adr1]) ∧ adr2 ∈ fs ∧ node-cons(fs[adr2].key, fs[adr2]) ∧ adr3 ∈ fs ∧ node-cons(fs[adr3].key, fs[adr3]) ∧ ro0.inodes? → ⟨old_dent := mkdentry(str, m3); p_inode0 := mkinode(m, md, true, m0, m1, m2); p_inode1 := mkinode(m3, md0, true, 1, m5, m6); afs_rmdir#(; p_inode0, p_inode1, old_dent, err0, dirs, files; asynced)⟩ ( dirs = abs-dirs(fs, ri) ∧ files = abs-files(fs, ri) ∧ orphans(abs-dirs(fs, ri), abs-files(fs, ri)) = abs-keys(ro0) ∧ (asynced → boolvar) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = old_dent ∧ ESUCCESS = err0)), fs0[ri0[dentrykey(m, str)]].ino = ROOT_INO, fs0[ri0[dentrykey(m, str)]].ino = 0, fs0[ri0[inodekey(m)]].nsubdirs = 0, fs0[ri0[inodekey(m)]].size = 0, adr1 = adr2, adr1 = adr3, adr2 = adr3, m = 0, adr3 ∈ fs0, adr2 ∈ fs0, adr1 ∈ fs0, fs0[ri0[inodekey(m)]].nlink = 0 ∧ m ≠ ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ #(∅ ++ key) = 1; (s)%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat%ubifs-parent-in: ⊦ dentrykey(ino, str) ∈ ri ∧ ubifs-cons(fs, ri, ro, fi, fo, log) → inodekey(ino) ∈ ri; (forward) from specification aubifs-inv%one-size: ⊦ ks = ∅ ++ key → # ks = 1; (forward) from specification set-basic[node-index]% ⊦ dentrykey(m, str).dentry?; (s)% ⊦ node-cons(key0, dentrynode(key1, ino)) ↔ key0.dentry?; (s)% ⊦ dentrynode(key, m).key = key; (s)% ⊦ adr ∈ fs[adr1, nd] ↔ adr = adr1 ∨ adr ∈ fs; (s)% ⊦ inodekey(m).ino = m; (s)% ⊦ inodekey(m).inode?; (s)%⊦ node-cons(key0, inodenode(key1, md, boolvar, n, n0, m)) ↔ key0.inode? ∧ key0.ino ≠ 0; (s)% ⊦ inodenode(key, md, boolvar, m, m0, m1).key = key; (s)% ⊦ fs[adr0, nd][adr1] = (adr0 = adr1 ⊃ nd;fs[adr1]); (s)% ⊦ adr ≠ adr0 → fs[adr0, nd][adr] = fs[adr]; (s)% ⊦ adr ∈ fs[adr0, nd] ↔ adr = adr0 ∨ adr ∈ fs; (ws)% ⊦ adr ≠ adr0 → (adr ∈ fs[adr0, nd] ↔ adr ∈ fs); (s)