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Proof step 225 from proof refine-rmdir (previous step)

(m = m4 ⊃ mkdir(m5, md0, m2, m7, entries);mkdir(m1, md, m3, m0, abs-children(m, fs0, ri0) -- str)) = mkdir(m6, md4, m8, m9, entries0), (m4 = m ⊃ mkdir(m1, md1, m3, m0, abs-children(m, fs0, ri0) -- str);mkdir(m15, md6, 1, m14, ∅)) = mkdir(m5, md3, m2 + 1, m7, entries), fs0[ri0[inodekey(m)]] = inodenode(key0, md1, true, m3, m0 + 1, m1 + 1), fs0[ri0[dentrykey(m, str)]] = dentrynode(key, m4), fs0[ri0[inodekey(m4)]] = inodenode(key2, md6, true, 1, m14, m15), abs-children(m4, fs0, ri0) = ∅, dentrykeys(inodekey(m4), ri0) = ∅, links(m4, fs0, ri0) = ∅ ++ dentrykey(m, str), m4 = m, mkdir(m5, md, m2, m7, entries) = mkdir(m10, md5, m11, m12, entries1), fo.inodes?, orphans(fs0, ri0).inodes?, afs-cons(abs-dirs(fs0, ri0), abs-files(fs0, ri0)), afs-cons (abs-dirs(fs0, ri0)[m, (m = m4 ⊃ mkdir(m5, md3, m2, m7, entries);mkdir(m1, md1, m3, m0, abs-children(m, fs0, ri0) -- str)).meta:= md] [m4, mkdir(m10, md0, m11, m12, entries1) ], 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(m4), md0, true, 0, m14, m15) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m4), 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(m4), md0, true, 0, m14, m15) ]), 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(m4), md0, true, 0, m14, m15) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m4), adr3], orphans(fs0, ri0) ++ inodekey(m4)), 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(m4), md0, true, 0, m14, m15) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m4), adr3], orphans(fs0, ri0) ++ inodekey(m4), 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(m4), 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(m4), md0, true, 0, m14, m15) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m4), adr3], orphans(fs0, ri0) ++ inodekey(m4), fi, fo, ((log0 + adr1 ') + adr2 ') + adr3 '), ubifs-cons(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), inodekey(ino) ∈ ri0, dentrykey(m, str) ∈ ri0, ri0[dentrykey(m, str)] ∈ fs0, inodekey(m4) ∈ ri0, ri0[inodekey(m4)] ∈ fs0, ri0[inodekey(m)] ∈ fs0, inodekey(m) ∈ ri0, asynced0 → boolvar0, m3 ≠ 0 ∨ m = ROOT_INO, links(m, abs-dirs(fs0, ri0)) ≠ ∅ ∨ m = ROOT_INO ⊦ adr1 = adr2, adr1 = adr3, adr2 = adr3, ino = m4, m = 0, m4 = ROOT_INO, m4 = 0, adr3 ∈ fs0, adr2 ∈ fs0, adr1 ∈ fs0, ( fs0[ri0[inodekey(ino)]].directory ↔ inodekey(ino) ∈ (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m4), adr3] ∧ (adr3 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m4), md0, true, 0, m14, m15) ;(adr2 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m), md, true, m3, m0, m1);fs0[(m = ino ⊃ adr1;ri0[inodekey(ino)])]) ) ) .directory) ∧ ( fs0[ri0[inodekey(ino)]].directory → (ino = m ⊃ mkdir(m1, md, m3, m0, abs-children(m, fs0, ri0) -- str) ;mkdir (fs0[ri0[inodekey(ino)]].size, fs0[ri0[inodekey(ino)]].meta, fs0[ri0[inodekey(ino)]].nlink, fs0[ri0[inodekey(ino)]].nsubdirs, abs-children(ino, fs0, ri0)) ) = mkdir ((adr3 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m4), md0, true, 0, m14, m15) ;(adr2 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m), md, true, m3, m0, m1);fs0[(m = ino ⊃ adr1;ri0[inodekey(ino)])]) ) ) .size, (adr3 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m4), md0, true, 0, m14, m15) ;(adr2 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m), md, true, m3, m0, m1);fs0[(m = ino ⊃ adr1;ri0[inodekey(ino)])]) ) ) .meta, (adr3 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m4), md0, true, 0, m14, m15) ;(adr2 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m), md, true, m3, m0, m1);fs0[(m = ino ⊃ adr1;ri0[inodekey(ino)])]) ) ) .nlink, (adr3 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m4), md0, true, 0, m14, m15) ;(adr2 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = (m = ino ⊃ adr1;ri0[inodekey(ino)]) ⊃ inodenode(inodekey(m), md, true, m3, m0, m1);fs0[(m = ino ⊃ adr1;ri0[inodekey(ino)])]) ) ) .nsubdirs, abs-children (ino, fs0[adr1, inodenode(inodekey(m), md, true, m3, m0, m1)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m4), md0, true, 0, m14, m15) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m4), adr3]))), m3 = 0 ∧ m ≠ ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 226:
fs0[ri0[dentrykey(m, str)]] = dentrynode(key, m), fs0[ri0[inodekey(m)]] = inodenode(key2, md6, true, 1, m12 + 1, m10 + 1), abs-children(m, fs0, ri0) = ∅, dentrykeys(inodekey(m), ri0) = ∅, links(m, fs0, ri0) = ∅ ++ dentrykey(m, str), orphans(fs0, ri0).inodes?, fo.inodes?, afs-cons(abs-dirs(fs0, ri0)[m, mkdir(m10, md0, 0, m12, ∅)], abs-files(fs0, ri0)), 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, 1, m12, m10)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m), adr3]), log-cons(log0, fs0), log-cons (log0, fs0[adr1, inodenode(inodekey(m), md, true, 1, m12, m10)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ]), orphans-cons(fs0, ri0, orphans(fs0, ri0)), orphans-cons (fs0[adr1, inodenode(inodekey(m), md, true, 1, m12, m10)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m), adr3], orphans(fs0, ri0) ++ inodekey(m)), replay-cons-strong(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), replay-cons-strong (fs0[adr1, inodenode(inodekey(m), md, true, 1, m12, m10)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m), adr3], orphans(fs0, ri0) ++ inodekey(m), 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(m), adr3]), ubifs-cons(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), ubifs-cons (fs0[adr1, inodenode(inodekey(m), md, true, 1, m12, m10)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m), adr3], orphans(fs0, ri0) ++ inodekey(m), fi, fo, ((log0 + adr1 ') + adr2 ') + adr3 '), ri0[inodekey(m)] ∈ fs0, inodekey(m) ∈ ri0, ri0[dentrykey(m, str)] ∈ fs0, dentrykey(m, str) ∈ ri0, inodekey(ino) ∈ ri0, asynced0 → boolvar0 ⊦ # links(m, abs-dirs(fs0, ri0)) = 0, adr1 = adr3, adr1 = adr2, adr2 = adr3, ino = m, links(m, abs-dirs(fs0, ri0)) = ∅, m = ROOT_INO, m = 0, adr1 ∈ fs0, adr2 ∈ fs0, adr3 ∈ fs0, ( fs0[ri0[inodekey(ino)]].directory ↔ inodekey(ino) ∈ (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m), adr3] ∧ (adr3 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ;(adr2 = ri0[inodekey(ino)] ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md, true, 1, m12, m10);fs0[ri0[inodekey(ino)]]) ) ) .directory) ∧ ( fs0[ri0[inodekey(ino)]].directory → mkdir (fs0[ri0[inodekey(ino)]].size, fs0[ri0[inodekey(ino)]].meta, fs0[ri0[inodekey(ino)]].nlink, fs0[ri0[inodekey(ino)]].nsubdirs, abs-children(ino, fs0, ri0)) = mkdir ((adr3 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ;(adr2 = ri0[inodekey(ino)] ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md, true, 1, m12, m10);fs0[ri0[inodekey(ino)]]) ) ) .size, (adr3 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ;(adr2 = ri0[inodekey(ino)] ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md, true, 1, m12, m10);fs0[ri0[inodekey(ino)]]) ) ) .meta, (adr3 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ;(adr2 = ri0[inodekey(ino)] ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md, true, 1, m12, m10);fs0[ri0[inodekey(ino)]]) ) ) .nlink, (adr3 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ;(adr2 = ri0[inodekey(ino)] ⊃ dentrynode(dentrykey(m, str), 0) ;(adr1 = ri0[inodekey(ino)] ⊃ inodenode(inodekey(m), md, true, 1, m12, m10);fs0[ri0[inodekey(ino)]]) ) ) .nsubdirs, abs-children (ino, fs0[adr1, inodenode(inodekey(m), md, true, 1, m12, m10)][adr2, dentrynode(dentrykey(m, str), 0)] [adr3, inodenode(inodekey(m), md0, true, 0, m12 + 1, m10 + 1) ], (ri0[inodekey(m), adr1] -- dentrykey(m, str))[inodekey(m), adr3])))

Rule argument

Additional information

Used simplifier rules: ⊦ #(∅ ++ key) = 1; (s)%size-zero: ⊦ # ls ≠ 0 → ls ≠ ∅; (forward) from specification set-basic[alink-set]% ⊦ ∅ -- str = ∅; (s)% ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ 1 = m + 1 ↔ m = 0; (s)%⊦ inodenode(key, md, boolvar, m, m0, m1) = inodenode(key0, md0, boolvar0, m2, m3, m4) ↔ key = key0 ∧ md = md0 ∧ (boolvar ↔ boolvar0) ∧ m = m2 ∧ m0 = m3 ∧ m1 = m4; (ws)%size-zero-01: ⊦ ls ≠ ∅ → # ls ≠ 0; (forward) from specification set-basic[alink-set]%one-size: ⊦ ks = ∅ ++ key → # ks = 1; (forward) from specification set-basic[node-index]% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)%⊦ mkdir(m, md, m0, m1, entries).meta:= md0 = mkdir(m, md0, m0, m1, entries); (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)