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Proof step 18 from proof coinduction (previous step)

φ(x), monotone(f), ν(f)(x0), ∀ x. φ(x) → f(λ x. ν(f)(x) ∨ φ(x))(x), ν(f) ⊆ (λ x. ν(f)(x) ∨ φ(x)) → f(ν(f)) ⊆ f(λ x. ν(f)(x) ∨ φ(x)) ⊦ f(λ x. ν(f)(x) ∨ φ(x))(x0), φ(x0)

User applied rule case distinction

And obtained the following new goals

Proof step 19:
φ(x), monotone(f), ν(f)(x0), ∀ x. φ(x) → f(λ x. ν(f)(x) ∨ φ(x))(x) ⊦ ν(f) ⊆ (λ x. ν(f)(x) ∨ φ(x)), f(λ x. ν(f)(x) ∨ φ(x))(x0), φ(x0)
Proof step 21:
f(ν(f)) ⊆ f(λ x. ν(f)(x) ∨ φ(x)), 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

pos: left 5

Proof step 18 from proof coinduction (previous step)

User applied rule case distinction

And obtained the following new goals

Rule argument

Additional information