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Proof step 14 from proof induction (previous step)

f(λ x. μ(f)(x) ∧ φ(x))(x0), f(μ(f)) ⊆ μ(f), μ(f)(x), monotone(f), φ(x0), ∀ x. f(λ x. μ(f)(x) ∧ φ(x))(x) → φ(x) ⊦ φ(x), μ(f)(x0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ S ⊆ S0 ↔ (∀ x. S(x) → S0(x))
Proof step 16:
f(λ x. μ(f)(x) ∧ φ(x))(x0), ∀ x. f(μ(f))(x) → μ(f)(x), μ(f)(x), monotone(f), φ(x0), ∀ x. f(λ x. μ(f)(x) ∧ φ(x))(x) → φ(x) ⊦ φ(x), μ(f)(x0)

Rule argument

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

Additional information

Used terms: μ(f), f(μ(f))