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Proof step 9 from proof afs-cons-reduce (previous step)

files1[ino] = mkfile(m1, md0, m2, content0), files0[ino] = mkfile(m, md, m0, content), afs-cons(dirs1, files1), afs-cons(dirs0, files0), ROOT_INO ∈ dirs0, ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅, ino ∈ dirs0 → dirs0[ino].nlink = # links(ino, dirs0) ∧ dirs0[ino].nlink ≤ 1 ∧ dirs0[ino].size = # dirs0[ino].entries ∧ dirs0[ino].nsubdirs = # subdirs(ino, dirs0), ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m0 = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files0 ⊦ (ino ∈ dirs1 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs1) = ∅ → dirs1[ino].entries = ∅) ∧ ( ino ∈ dirs1 → dirs1[ino].nlink = # links(ino, dirs1) ∧ dirs1[ino].nlink ≤ 1 ∧ dirs1[ino].size = # dirs1[ino].entries ∧ dirs1[ino].nsubdirs = # subdirs(ino, dirs1)) ∧ (ino ∈ files1 → ¬ ino ∈ dirs1 ∧ m2 = # links(ino, dirs1)) ∧ (ino ∈ dirs1 ∧ name ∈ dirs1[ino].entries → dirs1[ino].entries[name] ∈ dirs1 ∨ dirs1[ino].entries[name] ∈ files1)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
files0[ino] = mkfile(m, md, m0, content), files1[ino] = mkfile(m1, md0, m2, content0), dirs0[ino] = mkdir(m3, md1, m4, m5, entries), afs-cons(dirs0, files0), afs-cons(dirs1, files1), ROOT_INO ∈ dirs0, ino ∈ dirs0 ∧ name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files0, ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m0 = # links(ino, dirs0), ino ∈ dirs0 → m4 = # links(ino, dirs0) ∧ m4 ≤ 1 ∧ m3 = # entries ∧ m5 = # subdirs(ino, dirs0), ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries = ∅ ⊦ (ino ∈ dirs1 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs1) = ∅ → dirs1[ino].entries = ∅) ∧ ( ino ∈ dirs1 → dirs1[ino].nlink = # links(ino, dirs1) ∧ dirs1[ino].nlink ≤ 1 ∧ dirs1[ino].size = # dirs1[ino].entries ∧ dirs1[ino].nsubdirs = # subdirs(ino, dirs1)) ∧ (ino ∈ files1 → ¬ ino ∈ dirs1 ∧ m2 = # links(ino, dirs1)) ∧ (ino ∈ dirs1 ∧ name ∈ dirs1[ino].entries → dirs1[ino].entries[name] ∈ dirs1 ∨ dirs1[ino].entries[name] ∈ files1)

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] → [dirs0[ino], m3, md1, m4, m5, 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: dirs0[ino], m3, md1, m4, m5, entries