08eb78a5b2
This PR sets up the new integrated test/bench suite. It then migrates all benchmarks and some related tests to the new suite. There's also some documentation and some linting. For now, a lot of the old tests are left alone so this PR doesn't become even larger than it already is. Eventually, all tests should be migrated to the new suite though so there isn't a confusing mix of two systems.
55 lines
1.6 KiB
Text
55 lines
1.6 KiB
Text
namespace MWE
|
||
|
||
universe u v w
|
||
|
||
inductive Id {A : Type u} : A → A → Type u
|
||
| refl {a : A} : Id a a
|
||
|
||
infix:50 (priority := high) " = " => Id
|
||
|
||
def contr (A : Type u) := Σ (a : A), ∀ b, a = b
|
||
|
||
def singl {A : Type u} (a : A) :=
|
||
Σ b, a = b
|
||
|
||
def Corr (A : Type u) (B : Type v) :=
|
||
Σ (R : A → B → Type w), (∀ a, contr (Σ b, R a b)) × (∀ b, contr (Σ a, R a b))
|
||
|
||
def Homotopy {A : Type u} {B : A → Type v} (f g : ∀ x, B x) :=
|
||
∀ (x : A), f x = g x
|
||
infix:80 " ~ " => Homotopy
|
||
|
||
def isQinv {A : Type u} {B : Type v} (f : A → B) (g : B → A) :=
|
||
(f ∘ g ~ id) × (g ∘ f ~ id)
|
||
|
||
def Qinv {A : Type u} {B : Type v} (f : A → B) :=
|
||
Σ (g : B → A), isQinv f g
|
||
|
||
def Qinv.eqv (A : Type u) (B : Type v) :=
|
||
Σ (f : A → B), Qinv f
|
||
|
||
def linv {A : Type u} {B : Type v} (f : A → B) :=
|
||
Σ (g : B → A), g ∘ f ~ id
|
||
|
||
def rinv {A : Type u} {B : Type v} (f : A → B) :=
|
||
Σ (g : B → A), f ∘ g ~ id
|
||
|
||
def biinv {A : Type u} {B : Type v} (f : A → B) :=
|
||
linv f × rinv f
|
||
|
||
def Equiv (A : Type u) (B : Type v) : Type (max u v) :=
|
||
Σ (f : A → B), biinv f
|
||
infix:25 " ≃ " => Equiv
|
||
|
||
def ax₁ {A : Type u} {B : Type v} : A ≃ B → contr A → contr B :=
|
||
sorry
|
||
def ax₂ {A : Type v} {B : Type u} (f g : A → B) : (Σ x, f x = g x) ≃ (Σ x, g x = f x) :=
|
||
sorry
|
||
def ax₄ {A : Type u} {B : Type v} (w : Qinv.eqv A B) (b : B) : contr (Σ a, b = w.1 a) :=
|
||
sorry
|
||
def ax₃ {A : Type u} (a : A) : contr (singl a) :=
|
||
sorry
|
||
|
||
def corrOfQinv {A : Type u} {B : Type v} : Qinv.eqv A B → Corr A B :=
|
||
by { intro w; exists (λ a b => b = w.1 a); apply Prod.mk <;> intro x;
|
||
apply ax₁; apply ax₂; apply ax₃; apply ax₄ }
|