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Proof step 10 from proof del-commute-dels (previous step)

key0 ∈ ri, key0.data? → inodekey(key0.ino) ≠ key, key0.data? → ¬ inodekey(key0.ino) ∈ ks ⊦ ((ri ⊖ key) ⊖ ks)[key0] = ((ri ⊖ ks) ⊖ key)[key0], key0 = key, key0 ∈ ks, key ∈ ks, key0 ∈ ri ⊖ key ∧ ¬ key0 ∈ ks ∧ ¬ key0.data?

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: in-del-ro: ⊦ key ∈ ri ⊖ key0 ↔ key ∈ ri ∧ key ≠ key0 ∧ (key.data? → inodekey(key.ino) ≠ key0);