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Proof step 22 from proof cont-iterate-fix (previous step)

monotone(f), ((f ^ (n0 + 1)))(λ x. false)(x), ∀ x. f(λ x0. ∃ n. ((f ^ n))(λ x. false)(x0))(x) ↔ (∃ n. f(((f ^ n))(λ x. false))(x)) ⊦ ∃ n. f(((f ^ n))(λ x. false))(x), f(λ x0. ∃ n. ((f ^ n))(λ x. false)(x0))(x)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ ((f ^ (n + 1)))(S) = f(((f ^ n))(S))
Proof step 24:
monotone(f), f(((f ^ n0))(λ x. false))(x), ∀ x. f(λ x0. ∃ n. ((f ^ n))(λ x. false)(x0))(x) ↔ (∃ n. f(((f ^ n))(λ x. false))(x)) ⊦ ∃ n. f(((f ^ n))(λ x. false))(x), f(λ x0. ∃ n. ((f ^ n))(λ x. false)(x0))(x)

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