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Proof step 7 from proof del-commute-dels (previous step)

key0 ∈ (ri ⊖ key) ⊖ ks ↔ key0 ∈ (ri ⊖ ks) ⊖ key ⊦ key ∈ ks, (key0 ∈ (ri ⊖ key) ⊖ ks ↔ key0 ∈ (ri ⊖ ks) ⊖ key) ∧ (key0 ∈ (ri ⊖ key) ⊖ ks → ((ri ⊖ key) ⊖ ks)[key0] = ((ri ⊖ ks) ⊖ key)[key0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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