首页Prove the following by deductive system（Using First-order Logic）: If x does not occur in ψ, then ∃x : S • (φ(x) ⇒ ψ) ⊨ (∀x : S • φ(x)) ⇒ ψ.

Prove the following by deductive system（Using First-order Logic）: If x does not occur in ψ, then ∃x : S • (φ(x) ⇒ ψ) ⊨ (∀x : S • φ(x)) ⇒ ψ.

时间: 2024-05-30 20:13:01 浏览: 105

Leader-following consensus of second-order agents with multiple time-varying delays

In this paper a leader-following consensus problem of second-order multi-agent systems with fixed and switching topologies as well as non-uniform time-varying delays is considered For the case of fixed topology a necessary and sufficient condition is obtained For the case of switching topology a sufficient condition is obtained under the assumption that the total period over which the leader is globally reachable is sufficiently large We not only prove that a consensus is reachable asymptoticall

1. Assume that x does not occur in ψ. 2. Let a be an arbitrary element of S such that φ(a) ⇒ ψ. 3. Assume ∀x : S • φ(x). 4. By Universal Instantiation, φ(a). 5. Using Modus Ponens with (2) and (4), we can conclude that ψ. 6. Therefore, (∀x : S • φ(x)) ⇒ ψ, as desired.

阅读全文