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Proof step 8 from proof del-commute-dels (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 10:
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?
Proof step 18:
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

Rule argument

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

Additional information

Used terms: key0, ks, ri ⊖ key