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Proof step 7 from proof cont-monotone (previous step)

S0 ⊆ S, ∀ Sn. chain(Sn) → (∀ x. f(λ x0. ∃ n. Sn(n)(x0))(x) ↔ (∃ n. f(Sn(n))(x))) ⊦ f(S0) ⊆ f(S)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ S ⊆ S0 ↔ (∀ x. S(x) → S0(x))
Proof step 9:
S0 ⊆ S, ∀ Sn. chain(Sn) → (∀ x. f(λ x0. ∃ n. Sn(n)(x0))(x) ↔ (∃ n. f(Sn(n))(x))) ⊦ ∀ x. f(S0)(x) → f(S)(x)

Rule argument

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

Additional information

Used terms: f(S), f(S0)