mathlib3 documentation

data.list.tfae

The Following Are Equivalent #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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

source
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
source
theorem list.tfae_nil  :
list.nil.tfae
source
theorem list.tfae_singleton (p : Prop) :
[p].tfae
source
theorem list.tfae_cons_of_mem {a b : Prop} {l : list Prop} (h : b ∈ l) :
(a :: l).tfae ↔ (a ↔ b) ∧ l.tfae
source
theorem list.tfae_cons_cons {a b : Prop} {l : list Prop} :
(a :: b :: l).tfae ↔ (a ↔ b) ∧ (b :: l).tfae
source
theorem list.tfae_of_forall (b : Prop) (l : list Prop) (h : ∀ (a : Prop), a ∈ l → (a ↔ b)) :
l.tfae
source
theorem list.tfae_of_cycle {a b : Prop} {l : list Prop} :
list.chain (λ (_x _y : Prop), _x → _y) a (b :: l) → (list.ilast' b l → a) → (a :: b :: l).tfae
source
theorem list.tfae.out {l : list Prop} (h : l.tfae) (n₁ n₂ : ℕ) {a b : Prop} (h₁ : l.nth n₁ = option.some a . "refl") (h₂ : l.nth n₂ = option.some b . "refl") :
a ↔ b