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Proof step 7 from proof abs-dirs-del-dentry (previous step)

fs[ri[inodekey(ino)]].directory, fs[ri[inodekey(ino0)]].directory, inodekey(ino) ∈ ri, inodekey(ino0) ∈ ri, dentrykey(ino0, str0) ∈ ri ⊦ abs-children(ino0, fs, ri -- dentrykey(ino, str))[str0] = fs[ri[dentrykey(ino0, str0)]].ino, dentrykey(ino0, str0) = dentrykey(ino, str), ino = 0, ino0 = ino

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: at-children seq: ⊦ dentrykey(ino, str) ∈ ri → abs-children(ino, fs, ri)[str] = fs[ri[dentrykey(ino, str)]].ino subst: [str, ri, fs, ino] → [str0, ri -- dentrykey(ino, str), fs, ino0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ key ≠ key0 → (ri -- key0)[key] = ri[key]; (s)% ⊦ key ∈ ri -- key0 ↔ key ≠ key0 ∧ key ∈ ri; (s) Used terms: str0, ri -- dentrykey(ino, str), fs, ino0