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Proof step 105 from proof inv-mkdir (previous step)

replay-log(log0, fs1, fi ⊖ fo) = ri1 ⊖ ro, fs1[ri1[inodekey(m)]].directory, fs1[ri1[inodekey(m)]].inode?, fo.inodes?, ro.inodes?, file-node-cons(fs1, ri1), log-cons (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)]), log-cons(log0, fs1), orphans-cons(fs1, ri1, ro), replay-cons-strong(fs1, ri1, ro, fi, fo, log0), replay-cons-strong (fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)], ri1[inodekey(m), adr1], ro, fi, fo, log0 + adr1 '), replay-safe(log0, fs1, fi ⊖ fo), ri-cons (ri1[inodekey(m), adr1], fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)]), ri-inj(fi), ri-inj(ri1), ubifs-cons(fs1, ri1, ro, fi, fo, log0), ri1[inodekey(m)] ∈ fs1, inodekey(m) ∈ ri1, key ∈ ri1 → ((inodekey(m) = key ⊃ adr1;ri1[key]) = adr1 ∨ (inodekey(m) = key ⊃ adr1;ri1[key]) ∈ fs1) ∧ key-cons (key, (adr1 = (inodekey(m) = key ⊃ adr1;ri1[key]) ⊃ inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1) ;fs1[(inodekey(m) = key ⊃ adr1;ri1[key])] )), key = dentrykey(m, str) ∨ key ∈ ri1, fs1[ri1[inodekey(m)]].nlink ≠ 0 ∨ m = ROOT_INO ⊦ adr1 = adr3, adr1 = adr2, adr2 = adr3, m = 0, n = 0, adr2 ∈ fs1, inodekey(m) ∈ ro, dentrykey(m, str) ∈ ri1, inodekey(n) ∈ fo, inodekey(n) ∈ ri1, adr1 ∈ fs1, adr3 ∈ fs1, replay-safe (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) ∧ ¬ adr1 ∈ replay-log (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) ∧ ¬ adr2 ∈ replay-log (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) [inodekey(m), adr1 ] ∧ ¬ adr3 ∈ replay-log (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) [inodekey(m), adr1 ] [dentrykey(m, str), adr2 ] ∧ replay-log (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) [inodekey(m), adr1 ] [dentrykey(m, str), adr2 ] [inodekey(n), adr3 ] = ri1[inodekey(m), adr1][dentrykey(m, str), adr2][inodekey(n), adr3] ⊖ ro, ( (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) = adr2 ∨ (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) = adr1 ∨ (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) ∈ fs1) ∧ key-cons (key, (adr2 = (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) ⊃ dentrynode(dentrykey(m, str), n) ;(adr1 = (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) ⊃ inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1) ;fs1[(dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key]))] ) ))

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Proof step 106:
replay-log(log0, fs1, fi ⊖ fo) = ri1 ⊖ ro, fs1[ri1[inodekey(m)]].directory, fs1[ri1[inodekey(m)]].inode?, ro.inodes?, fo.inodes?, file-node-cons(fs1, ri1), log-cons(log0, fs1), log-cons (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)]), orphans-cons(fs1, ri1, ro), replay-cons-strong (fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)], ri1[inodekey(m), adr1], ro, fi, fo, log0 + adr1 '), replay-cons-strong(fs1, ri1, ro, fi, fo, log0), replay-safe(log0, fs1, fi ⊖ fo), ri-cons (ri1[inodekey(m), adr1], fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)]), ri-inj(ri1), ri-inj(fi), ubifs-cons(fs1, ri1, ro, fi, fo, log0), inodekey(m) ∈ ri1, ri1[inodekey(m)] ∈ fs1, key ∈ ri1, key ∈ ri1 → ((inodekey(m) = key ⊃ adr1;ri1[key]) = adr1 ∨ (inodekey(m) = key ⊃ adr1;ri1[key]) ∈ fs1) ∧ key-cons (key, (adr1 = (inodekey(m) = key ⊃ adr1;ri1[key]) ⊃ inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1) ;fs1[(inodekey(m) = key ⊃ adr1;ri1[key])] )), fs1[ri1[inodekey(m)]].nlink ≠ 0 ∨ m = ROOT_INO, key = dentrykey(m, str) ∨ key ∈ ri1 ⊦ adr1 = adr2, adr1 = adr3, adr2 = adr3, m = 0, n = 0, adr3 ∈ fs1, adr1 ∈ fs1, inodekey(n) ∈ ri1, inodekey(n) ∈ fo, dentrykey(m, str) ∈ ri1, inodekey(m) ∈ ro, adr2 ∈ fs1, ( (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) = adr2 ∨ (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) = adr1 ∨ (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) ∈ fs1) ∧ key-cons (key, (adr2 = (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) ⊃ dentrynode(dentrykey(m, str), n) ;(adr1 = (dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key])) ⊃ inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1) ;fs1[(dentrykey(m, str) = key ⊃ adr2;(inodekey(m) = key ⊃ adr1;ri1[key]))] ) )), replay-safe (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) ∧ ¬ adr1 ∈ replay-log (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) ∧ ¬ adr2 ∈ replay-log (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) [inodekey(m), adr1 ] ∧ ¬ adr3 ∈ replay-log (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) [inodekey(m), adr1 ] [dentrykey(m, str), adr2 ] ∧ replay-log (log0, fs1[adr1, inodenode(inodekey(m), md0, true, fs1[ri1[inodekey(m)]].nlink, fs1[ri1[inodekey(m)]].nsubdirs + 1, fs1[ri1[inodekey(m)]].size + 1)] [adr2, dentrynode(dentrykey(m, str), n) ] [adr3, inodenode(inodekey(n), md, true, 1, 0, 0) ], fi ⊖ fo) [inodekey(m), adr1 ] [dentrykey(m, str), adr2 ] [inodekey(n), adr3 ] = ri1[inodekey(m), adr1][dentrykey(m, str), adr2][inodekey(n), adr3] ⊖ ro

Rule argument

fma: key ∈ ri1

Additional information

Used simplifier rules: ⊦ key-cons(dentrykey(ino, str), nd) ↔ nd.dentry? ∧ nd.key = dentrykey(ino, str); (s)% ⊦ dentrynode(key, m).dentry?; (s)% ⊦ dentrynode(key, m).key = key; (s) Used formulas: key ∈ ri1