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Proof step 145 from proof refine-rmdir (previous step)

fs0[ri0[inodekey(m)]] = inodenode(key0, md1, true, m3, m0 + 1, m1 + 1), fs0[ri0[dentrykey(m, str)]] = dentrynode(key, m2), fs0[ri0[inodekey(m2)]] = inodenode(key1, md2, true, 1, m5, m6), abs-children(m2, fs0, ri0) = ∅, dentrykeys(inodekey(m2), ri0) = ∅, links(m2, fs0, ri0) = ∅ ++ dentrykey(m, str), fo.inodes?, orphans(fs0, ri0).inodes?, afs-cons(abs-dirs(fs0, ri0), abs-files(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]), file-node-cons(fs0, ri0), 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)]), log-cons(log0, fs0), 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)), orphans-cons(fs0, ri0, orphans(fs0, ri0)), 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 '), replay-cons-strong(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), ri-cons(ri0, fs0), ri-inj((ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m2), adr3]), ri-inj(fi), ri-inj(ri0), 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 '), ubifs-cons(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), dentrykey(m, str) ∈ ri0, ri0[dentrykey(m, str)] ∈ fs0, inodekey(m2) ∈ ri0, ri0[inodekey(m2)] ∈ fs0, ri0[inodekey(m)] ∈ fs0, inodekey(m) ∈ ri0, asynced → boolvar0, m3 ≠ 0 ∨ m = ROOT_INO, links(m, abs-dirs(fs0, ri0)) ≠ ∅ ∨ m = ROOT_INO ⊦ adr1 = adr2, adr1 = adr3, adr2 = adr3, m = 0, m2 = ROOT_INO, m2 = 0, adr3 ∈ fs0, adr2 ∈ fs0, adr1 ∈ fs0, m3 = 0 ∧ m ≠ ROOT_INO, str ∈ abs-dirs(fs0, ri0)[m].entries ∧ m2 = abs-dirs(fs0, ri0)[m].entries[str] ∧ m3 = abs-dirs(fs0, ri0)[m].nlink ∧ m1 + 1 = abs-dirs(fs0, ri0)[m].size ∧ m0 + 1 = abs-dirs(fs0, ri0)[m].nsubdirs ∧ fs0[ri0[inodekey(abs-dirs(fs0, ri0)[m].entries[str])]].directory ∧ 1 = fs0[ri0[inodekey(abs-dirs(fs0, ri0)[m].entries[str])]].nlink ∧ m6 = fs0[ri0[inodekey(abs-dirs(fs0, ri0)[m].entries[str])]].size ∧ m5 = fs0[ri0[inodekey(abs-dirs(fs0, ri0)[m].entries[str])]].nsubdirs

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 146:
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-dirs(fs0, ri0)[m] = mkdir(m4, md3, m7, m8, entries), abs-children(m2, fs0, ri0) = ∅, dentrykeys(inodekey(m2), ri0) = ∅, links(m2, fs0, ri0) = ∅ ++ dentrykey(m, str), 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, 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)), 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 ⊦ adr1 = adr3, adr1 = adr2, adr2 = adr3, m = 0, m2 = 0, m2 = ROOT_INO, adr1 ∈ fs0, adr2 ∈ fs0, adr3 ∈ fs0, str ∈ entries ∧ m2 = entries[str] ∧ m3 = m7 ∧ m1 + 1 = m4 ∧ m0 + 1 = m8 ∧ fs0[ri0[inodekey(entries[str])]].directory ∧ 1 = fs0[ri0[inodekey(entries[str])]].nlink ∧ m6 = fs0[ri0[inodekey(entries[str])]].size ∧ m5 = fs0[ri0[inodekey(entries[str])]].nsubdirs, m3 = 0 ∧ m ≠ ROOT_INO

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [abs-dirs(fs0, ri0)[m], m4, md3, m7, m8, entries]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: abs-dirs(fs0, ri0)[m], m4, md3, m7, m8, entries