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Proof step 21 from proof coinduction (previous step)

f(ν(f)) ⊆ f(λ x. ν(f)(x) ∨ φ(x)), φ(x), monotone(f), ν(f)(x0), ∀ x. φ(x) → f(λ x. ν(f)(x) ∨ φ(x))(x) ⊦ f(λ x. ν(f)(x) ∨ φ(x))(x0), φ(x0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ S ⊆ S0 ↔ (∀ x. S(x) → S0(x))
Proof step 23:
∀ x. f(ν(f))(x) → f(λ x. ν(f)(x) ∨ φ(x))(x), φ(x), monotone(f), ν(f)(x0), ∀ x. φ(x) → f(λ x. ν(f)(x) ∨ φ(x))(x) ⊦ f(λ x. ν(f)(x) ∨ φ(x))(x0), φ(x0)

Rule argument

spec: monotone-set-fixpoint inst: name: subseteq-def seq: ⊦ S ⊆ S0 ↔ (∀ x. S(x) → S0(x)) subst: [S0, S] → [f(λ x. ν(f)(x) ∨ φ(x)), f(ν(f))] flags: rewrite

Additional information

Used terms: f(λ x. ν(f)(x) ∨ φ(x)), f(ν(f))