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Proof step 19 from proof refine-rmdir (previous step)

abs-children(fs[ri[dentrykey(m, str)]].ino, fs, ri) = ∅, fs[ri[inodekey(ROOT_INO)]].directory, fs[ri[inodekey(m)]].directory, fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].directory, fo.inodes?, orphans(fs, ri).inodes?, file-node-cons(fs, ri), log-cons(log, fs), orphans-cons(fs, ri, orphans(fs, ri)), replay-cons-strong(fs, ri, orphans(fs, ri), fi, fo, log), ri-inj(fi), ri-inj(ri), ubifs-cons(fs, ri, orphans(fs, ri), fi, fo, log), inodekey(ROOT_INO) ∈ ri, dentrykey(m, str) ∈ ri, inodekey(fs[ri[dentrykey(m, str)]].ino) ∈ ri, inodekey(m) ∈ ri, asynced → boolvar, inodekey(fs[ri[dentrykey(m, str)]].ino) ∈ ri ∧ fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].directory ∧ fs[ri[dentrykey(m, str)]].ino ≠ ROOT_INO ∧ links(fs[ri[dentrykey(m, str)]].ino, abs-dirs(fs, ri)) = ∅ → abs-children(fs[ri[dentrykey(m, str)]].ino, fs, ri) = ∅, inodekey(fs[ri[dentrykey(m, str)]].ino) ∈ ri ∧ fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].directory → fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].nlink = # links(fs[ri[dentrykey(m, str)]].ino, abs-dirs(fs, ri)) ∧ fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].nlink ≤ 1 ∧ fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].size = # abs-children(fs[ri[dentrykey(m, str)]].ino, fs, ri) ∧ fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].nsubdirs = # subdirs(fs[ri[dentrykey(m, str)]].ino, abs-dirs(fs, ri)), inodekey(fs[ri[dentrykey(m, str)]].ino) ∈ ri ∧ ¬ fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].directory → fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].nlink = # links(fs[ri[dentrykey(m, str)]].ino, abs-dirs(fs, ri)), ∀ ino. (inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory ∧ ino ≠ ROOT_INO ∧ links(ino, abs-dirs(fs, ri)) = ∅ → abs-children(ino, fs, ri) = ∅) ∧ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → fs[ri[inodekey(ino)]].nlink = # links(ino, abs-dirs(fs, ri)) ∧ fs[ri[inodekey(ino)]].nlink ≤ 1 ∧ fs[ri[inodekey(ino)]].size = # abs-children(ino, fs, ri) ∧ fs[ri[inodekey(ino)]].nsubdirs = # subdirs(ino, abs-dirs(fs, ri))) ∧ (inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory → fs[ri[inodekey(ino)]].nlink = # links(ino, abs-dirs(fs, ri))) ∧ (∀ name. inodekey(ino) ∈ ri ∧ dentrykey(ino, name) ∈ ri → inodekey(abs-dirs(fs, ri)[ino].entries[name]) ∈ ri ∧ fs[ri[inodekey(abs-dirs(fs, ri)[ino].entries[name])]].directory ∨ inodekey(abs-dirs(fs, ri)[ino].entries[name]) ∈ ri ∧ ¬ fs[ri[inodekey(abs-dirs(fs, ri)[ino].entries[name])]].directory), ∀ name. inodekey(fs[ri[dentrykey(m, str)]].ino) ∈ ri ∧ dentrykey(fs[ri[dentrykey(m, str)]].ino, name) ∈ ri → inodekey(abs-dirs(fs, ri)[fs[ri[dentrykey(m, str)]].ino].entries[name]) ∈ ri ∧ fs[ri[inodekey(abs-dirs(fs, ri)[fs[ri[dentrykey(m, str)]].ino].entries[name])]].directory ∨ inodekey(abs-dirs(fs, ri)[fs[ri[dentrykey(m, str)]].ino].entries[name]) ∈ ri ∧ ¬ fs[ri[inodekey(abs-dirs(fs, ri)[fs[ri[dentrykey(m, str)]].ino].entries[name])]].directory, links(m, abs-dirs(fs, ri)) ≠ ∅ ∨ m = ROOT_INO ⊦ fs[ri[dentrykey(m, str)]].ino = ROOT_INO, fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].nlink = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].nlink = 0, fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].nsubdirs = # subdirs(fs[ri[dentrykey(m, str)]].ino, abs-dirs(fs, ri)), fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].size = 0, abs-children(fs[ri[dentrykey(m, str)]].ino, fs, ri) = ∅, links(fs[ri[dentrykey(m, str)]].ino, abs-dirs(fs, ri)) = ∅, fs[ri[inodekey(fs[ri[dentrykey(m, str)]].ino)]].directory, fs[ri[inodekey(m)]].directory, fs[ri[inodekey(ROOT_INO)]].directory, orphans(fs, ri).inodes?, fo.inodes?, file-node-cons(fs, ri), log-cons(log, fs), orphans-cons(fs, ri, orphans(fs, ri)), replay-cons-strong(fs, ri, orphans(fs, ri), fi, fo, log), ri-inj(ri), ri-inj(fi), ubifs-cons(fs, ri, orphans(fs, ri), fi, fo, log), inodekey(m) ∈ ri, inodekey(fs[ri[dentrykey(m, str)]].ino) ∈ ri, dentrykey(m, str) ∈ ri, inodekey(ROOT_INO) ∈ ri, asynced → boolvar, ∀ name. dentrykey(fs[ri[dentrykey(m, str)]].ino, name) ∈ ri → inodekey(fs[ri[dentrykey(fs[ri[dentrykey(m, str)]].ino, name)]].ino) ∈ ri ∧ fs[ri[inodekey(fs[ri[dentrykey(fs[ri[dentrykey(m, str)]].ino, name)]].ino)]].directory ∨ inodekey(fs[ri[dentrykey(fs[ri[dentrykey(m, str)]].ino, name)]].ino) ∈ ri ∧ ¬ fs[ri[inodekey(fs[ri[dentrykey(fs[ri[dentrykey(m, str)]].ino, name)]].ino)]].directory, ∀ ino. (inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory ∧ ino ≠ ROOT_INO ∧ links(ino, abs-dirs(fs, ri)) = ∅ → abs-children(ino, fs, ri) = ∅) ∧ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → fs[ri[inodekey(ino)]].nlink = # links(ino, abs-dirs(fs, ri)) ∧ fs[ri[inodekey(ino)]].nlink ≤ 1 ∧ fs[ri[inodekey(ino)]].size = # abs-children(ino, fs, ri) ∧ fs[ri[inodekey(ino)]].nsubdirs = # subdirs(ino, abs-dirs(fs, ri))) ∧ (inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory → fs[ri[inodekey(ino)]].nlink = # links(ino, abs-dirs(fs, ri))) ∧ (∀ name. inodekey(ino) ∈ ri ∧ dentrykey(ino, name) ∈ ri → inodekey(abs-dirs(fs, ri)[ino].entries[name]) ∈ ri ∧ fs[ri[inodekey(abs-dirs(fs, ri)[ino].entries[name])]].directory ∨ inodekey(abs-dirs(fs, ri)[ino].entries[name]) ∈ ri ∧ ¬ fs[ri[inodekey(abs-dirs(fs, ri)[ino].entries[name])]].directory), links(m, abs-dirs(fs, ri)) ≠ ∅ ∨ m = ROOT_INO ⊦ fs[ri[dentrykey(m, str)]].ino = ROOT_INO

Rule argument

Additional information

Used simplifier rules: ⊦ # ∅ = 0; (s)% ⊦ # ∅ = 0; (s)%size-zero-02: ⊦ # ls = 0 → ls = ∅; (forward) from specification set-basic[alink-set]% ⊦ m < 1 ↔ m = 0; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)%⊦ dentrykey(ino, str) ∈ ri → abs-children(ino, fs, ri)[str] = fs[ri[dentrykey(ino, str)]].ino; (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)