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Proof step 4 from proof cont-monotone (previous step)

∀ Sn. chain(Sn) → (∀ x. f(λ x0. ∃ n. Sn(n)(x0))(x) ↔ (∃ n. f(Sn(n))(x))) ⊦ monotone(f)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ monotone(f) ↔ (∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0))
Proof step 6:
∀ Sn. chain(Sn) → (∀ x. f(λ x0. ∃ n. Sn(n)(x0))(x) ↔ (∃ n. f(Sn(n))(x))) ⊦ ∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0)

Rule argument

spec: monotone-set-fixpoint inst: name: monotone-def seq: ⊦ monotone(f) ↔ (∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0)) subst: [f] → [f] flags: rewrite

Additional information

Used terms: f