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Proof step 40 from proof refine-rmdir (previous step)

fs[ri[key]].ino = fs[ri[dentrykey(m, str)]].ino, fs[ri[inodekey(fs[ri[key]].ino)]].nlink = 1, fs[ri[inodekey(fs[ri[key]].ino)]].nsubdirs = # subdirs(fs[ri[key]].ino, abs-dirs(fs, ri)), fs[ri[inodekey(fs[ri[key]].ino)]].size = 0, abs-children(fs[ri[key]].ino, fs, ri) = ∅, key.dentry?, fs[ri[inodekey(ROOT_INO)]].directory, fs[ri[inodekey(fs[ri[key]].ino)]].directory, fs[ri[inodekey(m)]].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(ROOT_INO) ∈ ri, inodekey(m) ∈ ri, inodekey(fs[ri[key]].ino) ∈ ri, dentrykey(m, str) ∈ ri, key ∈ ri, asynced → boolvar, ∀ 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. dentrykey(fs[ri[key]].ino, name) ∈ ri → inodekey(fs[ri[dentrykey(fs[ri[key]].ino, name)]].ino) ∈ ri ∧ fs[ri[inodekey(fs[ri[dentrykey(fs[ri[key]].ino, name)]].ino)]].directory ∨ inodekey(fs[ri[dentrykey(fs[ri[key]].ino, name)]].ino) ∈ ri ∧ ¬ fs[ri[inodekey(fs[ri[dentrykey(fs[ri[key]].ino, name)]].ino)]].directory, ∀ alink0. inodekey(alink0.pino) ∈ ri ∧ fs[ri[inodekey(alink0.pino)]].directory ∧ dentrykey(alink0.pino, alink0.name) ∈ ri ∧ fs[ri[key]].ino = fs[ri[dentrykey(alink0.pino, alink0.name)]].ino ↔ alink0 = alink, links(m, abs-dirs(fs, ri)) ≠ ∅ ∨ m = ROOT_INO ⊦ fs[ri[key]].ino = ROOT_INO, key = dentrykey(m, str)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 41:
⊦ key.dentry? ↔ (∃ ino, str. key = dentrykey(ino, str))
Proof step 42:
fs[ri[key]].ino = fs[ri[dentrykey(m, str)]].ino, fs[ri[inodekey(fs[ri[key]].ino)]].nlink = 1, fs[ri[inodekey(fs[ri[key]].ino)]].nsubdirs = # subdirs(fs[ri[key]].ino, abs-dirs(fs, ri)), fs[ri[inodekey(fs[ri[key]].ino)]].size = 0, abs-children(fs[ri[key]].ino, fs, ri) = ∅, fs[ri[inodekey(m)]].directory, fs[ri[inodekey(fs[ri[key]].ino)]].directory, fs[ri[inodekey(ROOT_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), key ∈ ri, dentrykey(m, str) ∈ ri, inodekey(fs[ri[key]].ino) ∈ ri, inodekey(m) ∈ ri, inodekey(ROOT_INO) ∈ ri, asynced → boolvar, ∀ alink0. inodekey(alink0.pino) ∈ ri ∧ fs[ri[inodekey(alink0.pino)]].directory ∧ dentrykey(alink0.pino, alink0.name) ∈ ri ∧ fs[ri[key]].ino = fs[ri[dentrykey(alink0.pino, alink0.name)]].ino ↔ alink0 = alink, ∀ name. dentrykey(fs[ri[key]].ino, name) ∈ ri → inodekey(fs[ri[dentrykey(fs[ri[key]].ino, name)]].ino) ∈ ri ∧ fs[ri[inodekey(fs[ri[dentrykey(fs[ri[key]].ino, name)]].ino)]].directory ∨ inodekey(fs[ri[dentrykey(fs[ri[key]].ino, name)]].ino) ∈ ri ∧ ¬ fs[ri[inodekey(fs[ri[dentrykey(fs[ri[key]].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), ∃ ino, str. key = dentrykey(ino, str), links(m, abs-dirs(fs, ri)) ≠ ∅ ∨ m = ROOT_INO ⊦ fs[ri[key]].ino = ROOT_INO, key = dentrykey(m, str)

Rule argument

spec: aubifs-asm name: elim-dentry seq: ⊦ key.dentry? ↔ (∃ ino, str. key = dentrykey(ino, str)) subst: [key] → [key] flags: <all paths>

Additional information

Used terms: key