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Proof step 10 from proof iterate-subseq (previous step)

monotone(f), ∀ f. monotone(f) → ((f ^ n0))(λ x. false) ⊆ ((f ^ (n0 + 1)))(λ x. false) ⊦ ((f ^ (n0 + 1)))(λ x. false) ⊆ f(((f ^ (n0 + 1)))(λ x. false))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ ((f ^ (n + 1)))(S) = f(((f ^ n))(S))
Proof step 12:
monotone(f), ∀ f. monotone(f) → ((f ^ n0))(λ x. false) ⊆ ((f ^ (n0 + 1)))(λ x. false) ⊦ f(((f ^ n0))(λ x. false)) ⊆ f(f(((f ^ n0))(λ x. false)))

Rule argument

spec: inst: name: iterate-rec seq: ⊦ ((f ^ (n + 1)))(S) = f(((f ^ n))(S)) subst: [S, n, f] → [λ x. false, n0, f] flags: rewrite, current

Additional information

Used terms: λ x. false, n0, f