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Proof step 3 from proof abs-files-unused-adrs (previous step)

range ri ∩ as = ∅ ⊦ (inodekey(ino) ∈ ri → (¬ fs[ri[inodekey(ino)]].directory ↔ ¬ (fs \ as)[ri[inodekey(ino)]].directory)) ∧ ( inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory → mkfile(fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, abs-content(ino, fs, ri)) = mkfile ((fs \ as)[ri[inodekey(ino)]].size, (fs \ as)[ri[inodekey(ino)]].meta, (fs \ as)[ri[inodekey(ino)]].nlink, abs-content(ino, fs \ as, ri)))

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 4:
range ri ∩ as = ∅, ri[inodekey(ino)] ∈ as ⊦ (inodekey(ino) ∈ ri → (¬ fs[ri[inodekey(ino)]].directory ↔ ¬ (fs \ as)[ri[inodekey(ino)]].directory)) ∧ ( inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory → mkfile(fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, abs-content(ino, fs, ri)) = mkfile ((fs \ as)[ri[inodekey(ino)]].size, (fs \ as)[ri[inodekey(ino)]].meta, (fs \ as)[ri[inodekey(ino)]].nlink, abs-content(ino, fs \ as, ri)))
Proof step 13:
range ri ∩ as = ∅ ⊦ ri[inodekey(ino)] ∈ as, (inodekey(ino) ∈ ri → (¬ fs[ri[inodekey(ino)]].directory ↔ ¬ (fs \ as)[ri[inodekey(ino)]].directory)) ∧ ( inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory → mkfile(fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, abs-content(ino, fs, ri)) = mkfile ((fs \ as)[ri[inodekey(ino)]].size, (fs \ as)[ri[inodekey(ino)]].meta, (fs \ as)[ri[inodekey(ino)]].nlink, abs-content(ino, fs \ as, ri)))

Rule argument

fma: ri[inodekey(ino)] ∈ as

Additional information

Used formulas: ri[inodekey(ino)] ∈ as