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Proof step 18 from proof induction (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 19:
⊦ monotone(f) ↔ (∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0))
Proof step 20:
f(λ x. μ(f)(x) ∧ φ(x))(x0), μ(f)(x), ∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0), φ(x0), ∀ x. f(μ(f))(x) → μ(f)(x), ∀ x. f(λ x. μ(f)(x) ∧ φ(x))(x) → φ(x) ⊦ φ(x), μ(f)(x0), f(μ(f))(x0)

Rule argument

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

Additional information

Used terms: f