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Proof step 4 from proof inter-union-diff (previous step)

⊦ adr ∈ (as0 ∪ (as1 \ as)) ∩ as ↔ adr ∈ as0 ∩ as

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-union, inst: node-store name: Intersect seq: ⊦ adr ∈ as ∩ as0 ↔ adr ∈ as ∧ adr ∈ as0 subst: [as0, as, adr] → [as, as0 ∪ (as1 \ as), adr] flags: <all paths>

Additional information

Used simplifier rules: ⊦ adr ∈ as → (adr ∈ (as0 \ as) ∪ as1 ↔ adr ∈ as1); (s)%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]% ⊦ adr ∈ as ∩ as0 ↔ adr ∈ as ∧ adr ∈ as0; (ws)%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store] Used terms: as, as0 ∪ (as1 \ as), adr