The Following Are Equivalent

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

def List.TFAE (l : List Prop) : Prop

TFAE: The Following (propositions) Are Equivalent.

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

Equations

• l.TFAE = ∀ (x : Prop), x ∈ l → ∀ (y : Prop), y ∈ l → (x ↔ y)

theorem List.tfae_nil : [].TFAE

@[simp]

theorem List.tfae_singleton (p : Prop) : [p].TFAE

theorem List.tfae_cons_of_mem {a b : Prop} {l : List Prop} (h : b ∈ l) : (a :: l).TFAE ↔ (a ↔ b) ∧ l.TFAE

theorem List.tfae_cons_cons {a b : Prop} {l : List Prop} : (a :: b :: l).TFAE ↔ (a ↔ b) ∧ (b :: l).TFAE

@[simp]

theorem List.tfae_cons_self {a : Prop} {l : List Prop} : (a :: a :: l).TFAE ↔ (a :: l).TFAE

theorem List.tfae_of_forall (b : Prop) (l : List Prop) (h : ∀ (a : Prop), a ∈ l → (a ↔ b)) : l.TFAE

theorem List.tfae_of_cycle {a b : Prop} {l : List Prop} (h_chain : Chain (fun (x1 x2 : Prop) => x1 → x2) a (b :: l)) (h_last : l.getLastD b → a) : (a :: b :: l).TFAE

theorem List.TFAE.out {l : List Prop} (h : l.TFAE) (n₁ n₂ : Nat) {a b : Prop} (h₁ : l[n₁]? = some a := by rfl) (h₂ : l[n₂]? = some b := by rfl) : a ↔ b

theorem List.forall_tfae {α : Type u_1} (l : List (α → Prop)) (H : ∀ (a : α), (map (fun (p : α → Prop) => p a) l).TFAE) : (map (fun (p : α → Prop) => ∀ (a : α), p a) l).TFAE

If P₁ x ↔ ... ↔ Pₙ x for all x, then (∀ x, P₁ x) ↔ ... ↔ (∀ x, Pₙ x). Note: in concrete cases, Lean has trouble finding the list [P₁, ..., Pₙ] from the list [(∀ x, P₁ x), ..., (∀ x, Pₙ x)], but simply providing a list of underscores with the right length makes it happier.

Example:

example (P₁ P₂ P₃ : ℕ → Prop) (H : ∀ n, [P₁ n, P₂ n, P₃ n].TFAE) :
    [∀ n, P₁ n, ∀ n, P₂ n, ∀ n, P₃ n].TFAE :=
  forall_tfae [_, _, _] H

theorem List.exists_tfae {α : Type u_1} (l : List (α → Prop)) (H : ∀ (a : α), (map (fun (p : α → Prop) => p a) l).TFAE) : (map (fun (p : α → Prop) => ∃ (a : α), p a) l).TFAE

If P₁ x ↔ ... ↔ Pₙ x for all x, then (∃ x, P₁ x) ↔ ... ↔ (∃ x, Pₙ x). Note: in concrete cases, Lean has trouble finding the list [P₁, ..., Pₙ] from the list [(∃ x, P₁ x), ..., (∃ x, Pₙ x)], but simply providing a list of underscores with the right length makes it happier.

Example:

example (P₁ P₂ P₃ : ℕ → Prop) (H : ∀ n, [P₁ n, P₂ n, P₃ n].TFAE) :
    [∃ n, P₁ n, ∃ n, P₂ n, ∃ n, P₃ n].TFAE :=
  exists_tfae [_, _, _] H

theorem List.tfae_not_iff {l : List Prop} : (map Not l).TFAE ↔ l.TFAE

theorem List.TFAE.not {l : List Prop} : l.TFAE → (map Not l).TFAE

Alias of the reverse direction of List.tfae_not_iff.