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Proof step 7 from proof iterate-subseq (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ ((f ^ (n + 1)))(S) = f(((f ^ n))(S))
Proof step 9:
∀ f. monotone(f) → ((f ^ n0))(λ x. false) ⊆ ((f ^ (n0 + 1)))(λ x. false), monotone(f) ⊦ ((f ^ n0 +1))(λ x. false) ⊆ f(((f ^ n0 +1))(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 +1, f] flags: rewrite, current

Additional information

Used terms: λ x. false, n0 +1, f