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Proof step 14 from proof coinduction (previous step)

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 spec-lemma

And obtained the following new goals

Proof step 15:
⊦ monotone(f) ↔ (∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0))
Proof step 16:
monotone(f) ↔ (∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0)), 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: monotone-def seq: ⊦ monotone(f) ↔ (∀ S, S0. S ⊆ S0 → f(S) ⊆ f(S0)) subst: [f] → [f] flags: []

Additional information

Used terms: f