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Proof step 9 from proof cont-iterate-fix (previous step)

monotone(f) ⊦ (∃ n. ((f ^ n))(λ x. false)(x)) ↔ f(λ x0. ∃ n. ((f ^ n))(λ x. false)(x0))(x), chain(λ n. ((f ^ n))(λ x. false))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ chain(Sn) ↔ (∀ m, n. Sn(m) ⊆ Sn(n) ∨ Sn(n) ⊆ Sn(m))
Proof step 11:
monotone(f) ⊦ (∃ n. ((f ^ n))(λ x. false)(x)) ↔ f(λ x0. ∃ n. ((f ^ n))(λ x. false)(x0))(x), ∀ m, n. ((f ^ m))(λ x. false) ⊆ ((f ^ n))(λ x. false) ∨ ((f ^ n))(λ x. false) ⊆ ((f ^ m))(λ x. false)

Rule argument

spec: <toplevel> name: chain-def seq: ⊦ chain(Sn) ↔ (∀ m, n. Sn(m) ⊆ Sn(n) ∨ Sn(n) ⊆ Sn(m)) subst: [Sn] → [λ n. ((f ^ n))(λ x. false)] flags: <all paths>

Additional information

Used terms: λ n. ((f ^ n))(λ x. false)