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Proof step 9 from proof induction (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ S ⊆ S0 ↔ (∀ x. S(x) → S0(x))
Proof step 11:
f(μ(f)) ⊆ μ(f), μ(f)(x), monotone(f), ∀ x. f(λ x. μ(f)(x) ∧ φ(x))(x) → φ(x) ⊦ φ(x), ∀ x. f(λ x. μ(f)(x) ∧ φ(x))(x) → μ(f)(x) ∧ φ(x)

Rule argument

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

Additional information

Used terms: λ x. μ(f)(x) ∧ φ(x), f(λ x. μ(f)(x) ∧ φ(x))