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Proof step 18 from proof del-commute-dels (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
key0 ∈ ri ⊦ ri[key0] = ((ri ⊖ ks) ⊖ key)[key0], key0 = key, key0.data?, key0 ∈ ks, key ∈ ks

Rule argument

Additional information

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