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Proof step 2 from proof abs-dirs-del-file (previous step)

inodekey(ino) ∈ ri ⊦ abs-dirs(fs, ri -- inodekey(ino)) = abs-dirs(fs, ri), fs[ri[inodekey(ino)]].directory

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
fs[ri[inodekey(ino0)]].directory, inodekey(ino) ∈ ri, inodekey(ino0) ∈ ri, dentrykey(ino0, str) ∈ ri ⊦ abs-children(ino0, fs, ri -- inodekey(ino))[str] = fs[ri[dentrykey(ino0, str)]].ino, fs[ri[inodekey(ino)]].directory

Rule argument

Additional information

Used simplifier rules: D-ext: ⊦ dirs = abs-dirs(fs, ri) ↔ (∀ ino. ( ino ∈ dirs ↔ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory) ∧ (ino ∈ dirs → dirs[ino] = abs-dir(ino, fs, ri)));%in-dirs: ⊦ ino ∈ abs-dirs(fs, ri) ↔ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory;%at-dirs: ⊦ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → abs-dirs(fs, ri)[ino] = abs-dir(ino, fs, ri);%dir: ⊦ abs-dir(ino, fs, ri) = mkdir (fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, fs[ri[inodekey(ino)]].nsubdirs, abs-children(ino, fs, ri));%D-inj: ⊦ mkdir(n0, md0, m0, m1, entries0) = mkdir(n1, md1, m2, m3, entries1) ↔ ¬(md0 ≠ md1 ∨ n0 ≠ n1 ∨ m0 ≠ m2 ∨ m1 ≠ m3 ∨ entries0 ≠ entries1);%C-ext: ⊦ entries = abs-children(ino, fs, ri) ↔ (∀ str. (str ∈ entries ↔ dentrykey(ino, str) ∈ ri) ∧ ( str ∈ entries → entries[str] = fs[ri[dentrykey(ino, str)]].ino));%in-children: ⊦ str ∈ abs-children(ino, fs, ri) ↔ dentrykey(ino, str) ∈ ri;% ⊦ inodekey(m) ≠ dentrykey(m0, str); (s)%key ≠ key0 ⊦ (ri -- key0)[key] = ri[key]; (s)% ⊦ inodekey(m) = inodekey(m0) ↔ m = m0; (s)% ⊦ key ∈ ri -- key0 ↔ key ≠ key0 ∧ key ∈ ri; (s)