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Proof step 7 from proof abs-dirs-gc-copy (previous step)

# ax = # kx, fs[ri[inodekey(ino)]].directory, new(ax, fs), ri-cons(ri, fs), inodekey(ino) ∈ ri, kx.toSet ⊆ dom ri ⊦ dups(kx), dups(ax), (dentrykey(ino, str) ∈ ri → dentrykey(ino, str) ∈ ri[kx, ax]) ∧ (dentrykey(ino, str) ∈ ri[kx, ax] → fs[ax, fs[ri[kx]]][ri[kx, ax][dentrykey(ino, str)]].ino = fs[ri[dentrykey(ino, str)]].ino)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: aubifs-inv name: read-lookup-copy seq: # ax = # kx ⊦ new(ax, fs) ∧ ¬ dups(ax) ∧ ¬ dups(kx) ∧ ri-cons(ri, fs) ∧ kx.toSet ⊆ dom ri ∧ key ∈ ri → fs[ax, fs[ri[kx]]][ri[kx, ax][key]] = fs[ri[key]] subst: [key, kx, ri, ax, fs] → [dentrykey(ino, str), kx, ri, ax, fs] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # kx = # ax → (key ∈ ri[kx, ax] ↔ key ∈ ri ∨ key ∈ kx); (ws)% ⊦ kx.toSet ⊆ dom ri ∧ ¬ key ∈ ri → ¬ key ∈ kx; (s) Used terms: dentrykey(ino, str), kx, ri, ax, fs