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Proof step 19 from proof del-commute-dels (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: at-del-ro-nodata seq: ⊦ key ∈ ri ∧ key ≠ key0 ∧ ¬ key.data? → (ri ⊖ key0)[key] = ri[key] subst: [key, key0, ri] → [key0, key, ri ⊖ ks] flags: <all paths>

Additional information

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