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Proof step 4 from proof datakeys-set-del (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

Rule argument

spec: set-union, inst: node-index name: Difference 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 simplifier rules: datakeys-set: ⊦ key ∈ datakeys(ks, ri) ↔ key ∈ ri ∧ key.data? ∧ inodekey(key.ino) ∈ ks;%datakeys: ⊦ key ∈ datakeys(key0, ri) ↔ key ∈ ri ∧ key.data? ∧ key0.inode? ∧ key.ino = key0.ino;% ⊦ inodekey(m).inode?; (s)% ⊦ inodekey(m).ino = m; (s)%ne-01: ⊦ key ∈ ks → ks ≠ ∅; (forward) from specification set-basic[node-index]% ⊦ key.inode? → inodekey(key.ino) = key; (s) Used terms: datakeys(key, ri), datakeys(ks, ri), key0