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Proof step 4 from proof datakeys-set-add (previous step)

⊦ key0 ∈ ri ∧ key0.data? ∧ (inodekey(key0.ino) = key ∨ inodekey(key0.ino) ∈ ks) ↔ key0 ∈ datakeys(ks, ri) ∪ datakeys(key, ri)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ key ∈ ks ∪ ks0 ↔ key ∈ ks ∨ key ∈ ks0
Proof step 6:
⊦ key0 ∈ ri ∧ key0.data? ∧ (inodekey(key0.ino) = key ∨ inodekey(key0.ino) ∈ ks) ↔ key0 ∈ datakeys(ks, ri) ∨ key0 ∈ datakeys(key, ri)

Rule argument

spec: set-union, inst: node-index name: Union seq: ⊦ key ∈ ks ∪ ks0 ↔ key ∈ ks ∨ key ∈ ks0 subst: [ks0, ks, key] → [datakeys(key, ri), datakeys(ks, ri), key0] flags: <all paths>

Additional information

Used terms: datakeys(key, ri), datakeys(ks, ri), key0