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Proof step 14 from proof iterate-subseq-leq (previous step)

monotone(f), m0 ≤ n0, ((f ^ m0))(λ x. false) ⊆ ((f ^ n0))(λ x. false) ⊦ f(((f ^ m0))(λ x. false)) ⊆ f(((f ^ n0))(λ x. false))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ monotone(f) ↔ (∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0))
Proof step 16:
∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0), m0 ≤ n0, ((f ^ m0))(λ x. false) ⊆ ((f ^ n0))(λ x. false) ⊦ f(((f ^ m0))(λ x. false)) ⊆ f(((f ^ n0))(λ x. false))

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