# Documentation

Mathlib.Data.List.TFAE

# The Following Are Equivalent #

This file allows to state that all propositions in a list are equivalent. It is used by Mathlib.Tactic.Tfae. TFAE l means ∀ x ∈ l, ∀ y ∈ l, x ↔ y∀ x ∈ l, ∀ y ∈ l, x ↔ y∈ l, ∀ y ∈ l, x ↔ y∀ y ∈ l, x ↔ y∈ l, x ↔ y↔ y. This is equivalent to pairwise (↔) l↔) l.

def List.TFAE (l : ) :

TFAE: The Following (propositions) Are Equivalent.

The tfae_have and tfae_finish tactics can be useful in proofs with TFAE goals.

Equations
theorem List.tfae_cons_of_mem {a : Prop} {b : Prop} {l : } (h : b l) :
List.TFAE (a :: l) (a b)
theorem List.tfae_cons_cons {a : Prop} {b : Prop} {l : } :
List.TFAE (a :: b :: l) (a b) List.TFAE (b :: l)
theorem List.tfae_of_forall (b : Prop) (l : ) (h : ∀ (a : Prop), a l → (a b)) :
theorem List.tfae_of_cycle {a : Prop} {b : Prop} {l : } :
List.Chain (fun x x_1 => xx_1) a (b :: l)(a) → List.TFAE (a :: b :: l)
theorem List.TFAE.out {l : } (h : ) (n₁ : ) (n₂ : ) {a : Prop} {b : Prop} (h₁ : autoParam (List.get? l n₁ = some a) _auto✝) (h₂ : autoParam (List.get? l n₂ = some b) _auto✝) :
a b