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Proof step 81 from proof afs-cons-rename-dir-negdentry (previous step)

(m0 = m ⊃ mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str);mkdir(m8, md5, # links(m0, dirs0), m10, entries3)) = mkdir(m4, md3, m6, m7, entries0), dirs0[m] = mkdir(m1 + 1, md, # links(m, dirs0), m2 + 1, entries2), dirs0[m0] = mkdir(m8, md5, # links(m0, dirs0), m10, entries3), dirs0[entries2[str]] = mkdir(m3, md4, # links(entries2[str], dirs0), m5, entries1), afs-cons(dirs0, files), m ∈ dirs0, str ∈ entries2, entries2[str] ∈ dirs0, m0 ∈ dirs0, ROOT_INO ∈ dirs0, m0 = m → md1 = md0, 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 ∈ files → ¬ ino ∈ dirs0 ∧ files[ino].nlink = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files, ino ∈ files ∧ n ∈ files[ino].content → # files[ino].content[n] = VFS_PAGE_SIZE, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ str0 ∈ entries3, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str) ) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ]) = ∅ → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .nlink = # links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str) ) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ]) ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .nlink ≤ 1 ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .size = # (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .nsubdirs = # subdirs (ino, dirs0 [m, (m = m0 ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str) ) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ])) ∧ ( ino ∈ files → ¬ ino ∈ dirs0 ∧ files[ino].nlink = # links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str) ) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ])) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries [name ] ∈ dirs0 ∨ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries [name ] ∈ files)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 82:
(m0 = m ⊃ mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str);mkdir(m8, md5, # links(m0, dirs0), m10, entries3)) = mkdir(m4, md3, m6, m7, entries0), dirs0[entries2[str]] = mkdir(m3, md4, # links(entries2[str], dirs0), m5, entries1), dirs0[m0] = mkdir(m8, md5, # links(m0, dirs0), m10, entries3), dirs0[m] = mkdir(m1 + 1, md, # links(m, dirs0), m2 + 1, entries2), files[ino] = mkfile(m9, md2, m11, content), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, m0 ∈ dirs0, entries2[str] ∈ dirs0, str ∈ entries2, m ∈ dirs0, ino ∈ files ∧ n ∈ content → # content[n] = VFS_PAGE_SIZE, ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files, ino ∈ files → ¬ ino ∈ dirs0 ∧ m11 = # links(ino, dirs0), 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 ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅, m0 = m → md1 = md0, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ str0 ∈ entries3, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str) ) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ]) = ∅ → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .nlink = # links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str) ) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ]) ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .nlink ≤ 1 ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .size = # (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .nsubdirs = # subdirs (ino, dirs0 [m, (m = m0 ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str) ) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ])) ∧ ( ino ∈ files → ¬ ino ∈ dirs0 ∧ m11 = # links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m1, md, # links(m, dirs0), m2, entries2 -- str) ) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ])) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries [name ] ∈ dirs0 ∨ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries0[str0, entries2[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries0[str0, entries2[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m1, md0, # links(m, dirs0), m2, entries2 -- str);dirs0[ino]) ) .entries [name ] ∈ files)

Rule argument

spec: abstract-file name: elim seq: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content) subst: [af, m, md, m0, content] → [files[ino], m9, md2, m11, content]

Additional information

Used simplifier rules: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content); (elim) Used terms: files[ino], m9, md2, m11, content