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Proof step 4 from proof abs-dirs-deldatakeys-2 (previous step)

⊦ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory ↔ inodekey(ino) ∈ ri \ datakeys(key, n, ri) ∧ fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].directory) ∧ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → 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)) = mkdir (fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].size, fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].meta, fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].nlink, fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].nsubdirs, abs-children(ino, fs, ri \ datakeys(key, n, ri))))

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 5:
inodekey(ino) ∈ ri ⊦ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory ↔ inodekey(ino) ∈ ri \ datakeys(key, n, ri) ∧ fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].directory) ∧ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → 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)) = mkdir (fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].size, fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].meta, fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].nlink, fs[(ri \ datakeys(key, n, ri))[inodekey(ino)]].nsubdirs, abs-children(ino, fs, ri \ datakeys(key, n, ri))))

Rule argument

fma: inodekey(ino) ∈ ri

Additional information

Used simplifier rules: ⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks; (ws) Used formulas: inodekey(ino) ∈ ri