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Proof step 7 from proof induction (previous step)

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 spec-lemma

And obtained the following new goals

Proof step 8:
monotone(f) ⊦ f(μ(f)) ⊆ μ(f)
Proof step 9:
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))

Rule argument

spec: monotone-set-fixpoint inst: name: prefixpoint seq: monotone(f) ⊦ f(μ(f)) ⊆ μ(f) subst: [f] → [f] flags: []

Additional information

Used terms: f