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Proof step 5 from proof coinduction (previous step)

φ(x), monotone(f), ∀ x. φ(x) → f(λ x. ν(f)(x) ∨ φ(x))(x) ⊦ UN(λ S. S ⊆ f(S))(x)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ UN(SS)(x) ↔ (∃ S. SS(S) ∧ S(x))
Proof step 7:
φ(x), monotone(f), ∀ x. φ(x) → f(λ x. ν(f)(x) ∨ φ(x))(x) ⊦ ∃ S. S ⊆ f(S) ∧ S(x)

Rule argument

spec: monotone-set-fixpoint inst: name: union-def seq: ⊦ UN(SS)(x) ↔ (∃ S. SS(S) ∧ S(x)) subst: [x, SS] → [x, λ S. S ⊆ f(S)] flags: rewrite

Additional information

Used terms: x, λ S. S ⊆ f(S)